Evidence for the weak coupling scenario of the Peierls transition in the blue bronze

On the basis of first-principles DFT calculations the wave-vector and temperature dependencies of the Lindhard response function of the blue bronze ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$ have been calculated. The $k_F^I+k_F^{II}$ interband component of the response, which is responsible for the Peierls instability, has been quantitatively analyzed. It is found that (i) the electron-hole coherence length of this response determines the length scale of the experimental intrachain CDW correlations, and (ii) the intrachain $q_{\parallel }$ dependence of such a response also determines the shape of the Kohn anomaly experimentally measured. These findings provide compelling evidence that the Peierls transition of the blue bronze ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$ follows the weak electron-phonon coupling scenario in the adiabatic approximation, something that had not yet been proved on the basis of first-principles calculations for a real material. It is proposed that the CDW interchain coupling occurs through a Coulomb coupling between dipolar CDWs. The nature of the phonon mode leading to the dipolar nature of the CDWs is also discussed, and the relevance of these results to rationalize the CDW instabilities in other oxides and bronzes is pointed out. These findings are also contrasted with recent results for other CDW materials like chalcogenides and tellurides.

Figure 1

Schematic illustration of why for a 1D system with two partially filled bands with different Fermi velocities only certain interband nestings may be effective.

Figure 2

Crystal structure of the blue bronzes. The lattice vectors shown in (a) are those of the $C$-centered cell of Graham and Wadsley [18]. The three different types of ${\mathrm{MoO}}_6$ octahedra are shown with different colors. The elementary building block of the structure (i.e., a cluster of ten octahedra) and the octahedral chains it generates along the $b$ direction are shown in (b). An alternative description of the octahedral chains associated with the red dashed line in (a) and the hump octahedra is shown in (c).

Figure 3

DFT band structure (a) and density of states (DOS) (b) for ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$. In (a) the size of the blue, red, and green dots are proportional to the Mo I, Mo II, and Mo III character, respectively. $\mathrm{\Gamma }=(0,0,0)$, ${\mathrm{X}}^{\prime }=(1/2,0,0)$, ${\mathrm{Y}}^{\prime }=(0,1/2,0)$, and ${\mathrm{Z}}^{\prime }=(0,0,1/2)$ in units of the $a^{\prime }$*, $b^{\prime }$*, and $c^{\prime }$* reciprocal lattice vectors (see beginning of Sec. 3 and Fig. 4). In (b) the total DOS and Mo I, Mo II, and Mo III projected DOS are shown. The DOS is given in units of states per eV per unit cell of 10 Mo and per spin direction.

Figure 4

DFT Fermi surface for ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$: (a) Representation using the rhombohedral Brillouin zone and (b) view along the perpendicular to the ($a$*,$b$*) plane. In (b) $q$ is the interband nesting vector.

Figure 5

DFT Lindhard response function for ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$ at 10 K.

Figure 6

Scans of the Lindhard response function along the $\mathrm{\Gamma }\text{−}\mathrm{Y}$ chain direction for different temperatures. This figure clearly shows the individual responses of the three Fermi surface nesting processes (i), (ii), and (iii) defined in the text.

Figure 7

Thermal dependence of the inverse coherence lengths along the chain ($\parallel$) and the intralayer ($\perp$) directions. These quantities are compared (dashed dotted lines) with the inverse experimental CDW correlation lengths reported in Ref. [51].

Figure 8

Logarithmic dependence of the electron-hole response for the interband nesting process of the blue bronze according to Eq. (8).

Figure 9

Fitting of the square of the frequency Kohn anomaly measured at 230 K in Ref. [24] [see also Fig. 13] with the calculated Lindhard function. The best fit allows us to determine the base line corresponding to the square of the bare critical phonon mode ${\mathrm{\Omega }}_0^2\left(q\right)$.

Figure 10

Scans of the Lindhard response function along the intralayer transverse direction for different temperatures. Note that the maxima at about $0.12a$'* corresponds to $0a$*.

Figure 11

Lateral phasing of the dipolar CDWs in ${\mathrm{K}}_{0.3}{\mathrm{MoO}}_3$.

Figure 12

Illustration of the low-frequency phonon coupling mechanism for interchain coupling in the blue bronze and its detection. (a) Local polarization induced by the correlated Mo displacements in a segment made of corner sharing octahedra. (b) Planar diffuse scattering developed in x-ray scattering measurements due to the correlated Mo displacements shown in (a). (c) A sheet of low-frequency phonons perpendicular to the polarization direction.

Figure 13

(a) Dispersion of the low-lying phonon branches of the blue bronze measured between 225 and 295 K. The acoustic branches are labeled according to their polarization defined in the text. The hybridized phonon branches involved in the Peierls instability are colored in red and blue for the acoustic and optic counterparts, respectively. The empty green circles outline the phonon modes in the vicinity of the Kohn anomaly drawn at 230 K. (b) and (c): Sections of the Brillouin zone scanned during the inelastic scattering investigations (adapted from Refs. [21, 76]).