Unfolding First-Principles Band Structures

A general method is presented to unfold band structures of first-principles supercell calculations with proper spectral weight, allowing easier visualization of the electronic structure and the degree of broken translational symmetry. The resulting unfolded band structures contain additional rich information from the Kohn-Sham orbitals, and absorb the structure factor that makes them ideal for a direct comparison with angle resolved photoemission spectroscopy experiments. With negligible computational expense via the use of Wannier functions, this simple method has great practical value in the studies of a wide range of materials containing impurities, vacancies, lattice distortions, or spontaneous long-range orders.

Figure 1

Illustration of band folding in the supercell calculations: (a) band structure of a two-dimensional one-band first-neighbor tight-binding model, (b) the same obtained from a $4\times 4$ supercell calculation, and (c) the same obtained from a $16\times 16$ supercell calculation. Panel (d) shows the DOS.

Figure 2

Lattice structures of (a) ${\mathrm{Co}}_2{\mathrm{O}}_4$ (normal cell) and (b) ${\mathrm{Na}}_2{\mathrm{Co}}_6{\mathrm{O}}_{12}$ (supercell), the corresponding band structure of (c) the normal cell and (d) supercell calculation, and (e) the unfolded band structure of the supercell. The inset illustrates the effects of weak translational symmetry breaking via spectral functions over the region [$-4.6\mathrm{eV}$, $-4.2\mathrm{eV}$] and [$\frac{1}{5}\mathrm{M}\Gamma$, $\frac{3}{5}\mathrm{M}\Gamma$].

Figure 3

Lattice structures of (a) normal cell and (b) doubled-size supercell of $A$-type antiferromagnetic ordered ${\mathrm{LaMnO}}_3$, without and with the orbital ordering, respectively, the corresponding band structure of the normal cell (c) and the supercell (d), and the unfolded band structure of the supercell (e) indicating the orbital-ordering gap, ${\Delta }_{\mathrm{OO}}$. Red and blue bands denote the $z^2$ and $x^2-y^2$ orbital characters of the spin-majority channel, and green bands give both $e_g$ characters of the spin-minority channel.