Dispersion Relations of t2g bands of WO3 Lattice
As discussed in Chapter 3 of the manual, the hexagonal potassium tungsten bronze KxWO3 has a 3D WO3
lattice made up of corner-sharing WO6 octahedra, and the potassium cations K+ are located in the
hexagonal tunnels of the lattice. When all the K+ sites are occupied, x = 1/3. As the temperature is
lowered, KxWO3 undergoes a metal-to-semiconductor-to-metal phase transition in its electrical
resistivity along the crystallographic c-direction, but not along the directions perpendicular to the c direction.
To understand the electrical and superconducting properties of this compound,
electronic band structure calculations were carried out.
In the CAESAR\Examples\Chapter7\KxWO3 directory, the input files Dsp.BI and Dsp.PI are designed to obtain smooth plots of dispersion relations, while the input files Dos.BI and Dos.PI are designed to obtain the DOS and PDOS plots.
Since KxWO3 has the formula (KxWO3)6 per unit cell, the t2g-block bands of KxWO3 consist of 18 bands. Given a complete charge transfer from the potassium atom K to the 3D WO3 lattice, there are at most two electrons (i.e., 6x) per unit to fill the t2g-block bands. Therefore, only a few bands lying at the bottom of the t2g-block bands become partially filled. The following plot shows the dispersion relations of the bottom portion of these bands. For the case of x = 1/3, the band dispersion relations along G-Z reveal that the bottom three bands are partially filled. The band dispersion relations along G-X and G-M (i.e., in the a*b*-plane) show that the lowest-lying band does not cross the Fermi level, but the other two bands do. Consequently, the lowest-lying band is 1D, and the other two partially filled bands are 3D in nature. As the x value decreases from 1/3 to 0.18, the Fermi level is lowered and eventually cuts the lowest-lying band around point K = (1/3, 1/3, 0). For x > 0.18, therefore, this band keeps its 1D character, while the other two partially filled bands keep their 3D character.
In the CAESAR\Examples\Chapter7\KxWO3 directory, the input files Dsp.BI and Dsp.PI are designed to obtain smooth plots of dispersion relations, while the input files Dos.BI and Dos.PI are designed to obtain the DOS and PDOS plots.
Since KxWO3 has the formula (KxWO3)6 per unit cell, the t2g-block bands of KxWO3 consist of 18 bands. Given a complete charge transfer from the potassium atom K to the 3D WO3 lattice, there are at most two electrons (i.e., 6x) per unit to fill the t2g-block bands. Therefore, only a few bands lying at the bottom of the t2g-block bands become partially filled. The following plot shows the dispersion relations of the bottom portion of these bands. For the case of x = 1/3, the band dispersion relations along G-Z reveal that the bottom three bands are partially filled. The band dispersion relations along G-X and G-M (i.e., in the a*b*-plane) show that the lowest-lying band does not cross the Fermi level, but the other two bands do. Consequently, the lowest-lying band is 1D, and the other two partially filled bands are 3D in nature. As the x value decreases from 1/3 to 0.18, the Fermi level is lowered and eventually cuts the lowest-lying band around point K = (1/3, 1/3, 0). For x > 0.18, therefore, this band keeps its 1D character, while the other two partially filled bands keep their 3D character.
Figure. Dispersion relations for the bottom portion of the t2gblock bands of the 3D WO3 lattice of KxWO3. The dashed line refers to the Fermi level corresponding to x = 1/3.
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