Effective model for the electronic properties of quasi-one-dimensional purple bronze Li${}_{0.9}$Mo${}_6$O${}_{17}$ based on ab initio calculations

We investigate the electronic structure of the strongly anisotropic, quasi-low-dimensional purple bronze Li${}_{0.9}$Mo${}_6$O${}_{17}$. Building on all-electron ab initio band-structure calculations, we obtain an effective model in terms of four maximally localized Wannier orbitals, which turn out to be far from atomiclike. We find two half-filled orbitals arranged in chains running along one crystallographic direction and two full orbitals in perpendicular directions, respectively. The possibility to reduce this model to only two orbitals forming two chains per unit cell with interchain coupling is discussed. Transport properties of these models show high anisotropy, reproducing trends of the experimentally determined values for the dc conductivity. We also consider basic effects of electron-electron interactions using the (extended) variational cluster approach and dynamical mean field theory. We find good agreement with experimental photoemission data upon adding moderate onsite interaction of the order of the bandwidth to the ab initio derived tight-binding Hamiltonian. The obtained models provide a profound basis for further investigations on low-energy Luttinger-liquid properties or to study electronic correlations within computational many-body theory.

Figure 1

Left: LDA band structure (solid black) in the vicinity of the Fermi energy ${\epsilon }_F$ plotted along a two-dimensional path in the reciprocal $b$-$c$ plane. The four bands of the full Wannier projected Hamiltonian are shown on top (solid green) next to the data for the reduced model (dashed-dotted blue line). Right: DOS of the LDA calculation (black), the full Wannier model (solid green line), and the reduced model (dashed-dotted blue line).

Figure 2

Calculated Fermi surface. Left: The ab initio result (image created using xcrysden [59]). Center: In-plane projection of the result for the four-orbital Wannier model. Right: In-plane projection of the result for the reduced model, where type-A orbitals are strictly one dimensional and the two Fermi sheets are degenerate.

Figure 3

Top plot: Crystal structure. In the $a$ and $c$ directions one unit cell and in the $b$ direction two unit cells are shown. Big gray balls are Mo${}_1$/Mo${}_1^{\prime }$ and Mo${}_4$/Mo${}_4^{\prime }$ atoms, showing the zigzag chain structure along $b$. Small gray: other Mo sites. The numbers next to the black arrows denote the atom number. Small red balls are oxygen, and the yellow balls Li atoms. This image has been created using xcrysden [59]. Bottom plot: Partial DOS of the six inequivalent Mo atoms in the unit cell. Top row: atoms Mo${}_1$ and Mo${}_4$, forming the zigzag chains. Middle row: Mo${}_2$ and Mo${}_5$. Bottom row: Mo${}_3$ and Mo${}_6$.

Figure 4

Visualization of one type-A Wannier orbital. Similar as in Fig. 3, two unit cells in the $b$ direction are shown, the Wannier function is centered in the lower unit cell. The center of the Wannier function is located in-between atoms Mo${}_1$ and Mo${}_4^{\prime }$. Arrows mark the atoms with significant contribution to the Wannier function: in-chain atoms Mo${}_1$ and Mo${}_4^{\prime }$, and adjacent atoms Mo${}_2$ and Mo${}_5^{\prime }$. Color coding as in the top panel of Fig. 3. For the Wannier functions, blue and green lobes denote positive and negative phases, respectively. The image has been created using xcrysden [59].

Figure 5

Visualization of the full effective Wannier Hamiltonian (1). A schematic drawing of the most dominant hopping processes is presented: type-A orbitals (red), type-B orbitals (blue). Lines denote the dominant hopping paths, and the black square marks the size of the unit cell.

Figure 6

Spectral function and DOS of the interacting model. Top row: VCA data for $U=1$ eV using eight-orbital clusters. Bottom row: DMFT data for $U=1$ eV. Left: Spectral function plotted along a two-dimensional path in the reciprocal $b$-$c$ plane. The noninteracting dispersion is plotted on top (dashed green line). Center: Orbitally resolved density of states. Right: Zoom to the spectral function in the respective red rectangle compared to ARPES data from Ref. [22] which are indicated as red circles.

Figure 7

Cuts through the spectral function along the $b$ axis. The parameters and labels (I, II) correspond to those in Fig. 6.

Figure 8

Spectral function and DOS of the interacting model for high values of onsite interaction strength as one would expect for atomiclike molybdenum $d$ orbitals: $U=6$ eV. For legends, arrangement of the subplots, and color coding, see Fig. 6. On the far right we show cuts through the spectral function as in Fig. 7.

Figure 9

Comparison of VCA cluster sizes for a moderate onsite interaction strength of $U=0.5$ eV. Top row: VCA data for using four-orbital clusters. Bottom row: VCA data for using eight-orbital clusters. For legends, arrangement of the subplots, and color coding, see Fig. 8.