Charge density waves and electron-hole instabilities of the hidden-nesting materials $P_4{W}_{12}{O}_{44}, \gamma$- and $\eta \text{−}{\mathrm{Mo}}_4{O}_{11}$

The origin of the charge density wave (CDW) instabilities in the isostructural but not isoelectronic octahedral layers of the three-dimensional solids $\gamma \text{−}{\mathrm{Mo}}_4{O}_{11}$ and $P_4{W}_{12}{O}_{44}$ is discussed on the basis of first-principles density functional theory calculations. These layers contain three different and superposed one-dimensional (1D) systems (two diagonal and one horizontal) associated with the three $t_{2g}$ orbitals of the transition metal in octahedral coordination. Because of the special topology of the layers the three 1D systems are practically independent (hidden nesting) and the Lindhard function contains three different lines of intensity maxima associated with each of them. Clear cusps (six for $P_4{W}_{12}{O}_{44}$ and four for $\gamma \text{−}{\mathrm{Mo}}_4{O}_{11}$) occur at the intersections of these intensity lines. The wave vector of the structural modulations associated with some of these cusps from our calculations is in good agreement with the observed CDW wave vectors. The nature of the different modulations is analyzed on the basis of the calculated thermal dependence of intrachain and interchain coherence lengths of the diffuse lines associated with the diagonal and horizontal chains. Modulation in the diagonal chains is found to be more favorable than in the horizontal chain. The same type of wave vector is selected for $\gamma \text{−}{\mathrm{Mo}}_4{O}_{11}$ and $P_4{W}_{12}{O}_{44}$ despite having a different band filling. The coupling of the electronic instability to the phonon spectra and the relationship between the nature of the high-temperature modulation with the width of the octahedral layers is discussed. Among the two Magnéli phases the interlayer coupling is found to be somewhat stronger in $\eta \text{−}{\mathrm{Mo}}_4{O}_{11}$. The relationship with other hidden-nesting series of materials as the rare-earth tellurides is commented.