Calculation of model Hamiltonian parameters for ${\text{LaMnO}}_3$ using maximally localized Wannier functions

In this work we demonstrate that maximally localized Wannier functions (MLWFs) based on Kohn-Sham band structures provide a very robust and systematic way to construct realistic, materials-specific tight-binding models for further theoretical analysis. In particular, we construct MLWFs for the $\text{Mn}{e}_g$ bands in ${\text{LaMnO}}_3$, and we monitor changes in the MLWF matrix elements induced by different magnetic configurations and structural distortions. By comparing our results with commonly used model Hamiltonians for manganites, where electrons can hop between two “$e_g$-like” orbitals located on each Mn site, we obtain values for the local Jahn-Teller and Hund’s rule coupling strength, the hopping amplitudes between all nearest and further neighbors, and the corresponding reduction due to the ${\text{GdFeO}}_3$-type distortion. In addition, our analysis allows us to systematically assess and quantify the limitations of such an effective $e_g$-band description. We find that the most crucial limitation of such models stems from neglecting changes in the underlying $\text{Mn}\left(d\right)\text{-O}\left(p\right)$ hybridization, which not only lead to a significant difference in hopping for (local) spin majority/minority electrons but also to a nonlocal effect of the Jahn-Teller distortion and a significant reduction in the local Jahn-Teller coupling strength due to the ${\text{GdFeO}}_3$-type distortion.

Figure 1

(Color online) Different structural modifications of ${\text{LaMnO}}_3$ investigated in this work, viewed along the [001] direction: (i) ideal cubic perovskite, (ii) purely Jahn-Teller distorted, (iii) purely ${\text{GdFeO}}_3$-type distorted, and (v) experimental $Pbnm$ structure. Pictures have been generated using VESTA (Ref. 26).

Figure 2

(Color online) Projected DOS and band structure along high symmetry lines within the BZ calculated for the cubic structure (i) and both FM and A-AFM order. Filled (red) areas and solid (green) lines represent the projected DOS corresponding to $\text{Mn}\left(e_g\right)$ and $\text{Mn}\left(t_{2g}\right)$ states, respectively, while dashed (blue) lines represent the site and orbitally averaged projected DOS corresponding to the $\text{O}p$ states. For the A-AFM case the left (right) panel corresponds to local majority (minority) spin projection. In the band structure plots, the dispersion calculated from the $e_g$-like MLWFs are represented by thick (red) lines, whereas the thin (gray) lines represent the DFT band structure. The Fermi level is indicated by the horizontal dashed lines.

Figure 3

(Color online) Real-space representation of the MLWFs for majority and minority spin projections in the cubic structure (i) with FM order, projected on the $x\text{−}z$ plane passing through Mn (large/blue sphere) and O (small/red spheres) sites (in arbitrary units).

Figure 4

(Color online) Magnitude of all calculated nonzero hopping parameters for FM order in the ideal cubic structure as a function of the intersite distance $\left|\Delta \mathbf{R}\right|$ (open circles: hopping along the unit cell directions; open diamonds: hopping between next-nearest neighbors; filled circles: all other hoppings). Inset: comparison of the full MLWF band structure (solid lines) and the one calculated from a simplified TB model (filled circles) which includes only the inter-site hoppings for which the largest matrix element is larger than 10 meV (see main text).

Figure 5

(Color online) DFT band structure (thin lines) for the JT-distorted structure (ii): (a) majority spin FM, (b) minority spin FM, and (c) A-AFM. MLWF bands are depicted as thick/red lines. The Fermi level is indicated by the dashed line.

Figure 6

(Color online) MLWF Hamiltonian matrix elements $h_{ab}^{\Delta \mathbf{R}}$ as function of the JT distortion. Large/black and small/red symbols correspond to FM and A-AFM order, respectively. Matrix elements associated with pure (local) majority and minority spin character are shown as triangles pointing up and down, respectively. Closed circles in (b) represent the A-AFM $h_{11}^z$ hopping.

Figure 7

(Color online) DFT band structure (thin lines) for the purely GFO-distorted structure (iii): (a) FM majority spin, (b) FM minority spin, and (c) A-AFM. MLWF bands are depicted as thick/red lines. The Fermi level is indicated by dashed lines.

Figure 8

(Color online) Hamiltonian matrix elements in the basis of MLWFs as a function of the GFO distortion. Large/black and small/red symbols correspond to FM and A-AFM order, respectively. Elements associated with purely (local) majority and minority spin characters are represented by triangles pointing up and down, respectively.

Figure 9

(Color online) DFT band structure (thin lines) for structure (iv): (a) majority spin FM, (b) minority spin FM, and (c) A-AFM. MLWF are depicted as thick/red lines. Fermi level is indicated by dashed line.

Figure 10

(Color online) MLWF Hamiltonian matrix elements as function of combined JT and GFO distortion. Large/black and small/red symbols correspond to the FM and A-AFM order, respectively. Elements associated with purely (local) majority and minority spin characters are represented by triangles pointing up and down, respectively.

Figure 11

(Color online) Top: (a) DFT bands (thin/gray lines) and MLWF bands (thick/red lines) for the A-AFM experimental $Pbnm$ structure (v). (b) Comparison of the MLWF bands with the refined TB model and (c) with the simple nearest-neighbor TB model. Bottom: comparison between MLWF bands (thick/red lines) and refined TB model (thin lines with diamonds) for structures (i)–(iv) and A-AFM order. The Fermi level is indicated by dashed lines.

Figure 12

(Color online) Deviations $\Delta h_{ab}^{\Delta \mathbf{R}}$ of the selected matrix elements from the cubiclike form as a function of the transformation angle $\theta$. Vanishing deviations (and the corresponding angles ${\theta }_0$, ${\theta }_z$, and ${\theta }_x$) are marked by vertical dashed lines.