Structural instability and the Mott-Peierls transition in a half-metallic hollandite : ${\mathrm{K}}_2{\mathrm{Cr}}_8{\mathrm{O}}_{16}$

In order to explore the driving mechanism of the concomitant metal-insulator and structural transitions in quasi-one-dimensional hollandite ${\mathrm{K}}_2{\mathrm{Cr}}_8{\mathrm{O}}_{16}$, electronic structures and phonon properties are investigated by employing the ab initio density functional theory (DFT) calculations. We have found that the imaginary phonon frequency reflecting the structural instability appears only in the DFT + $U$ ($U$: Coulomb correlation) calculation, which indicates that the Coulomb correlation plays an essential role in the structural transition. The lattice displacements of the softened phonon at $X$ explain the observed lattice distortions in ${\mathrm{K}}_2{\mathrm{Cr}}_8{\mathrm{O}}_{16}$ perfectly well, suggesting the Peierls distortion vector Q of $X$ (0, 0, 1/2). The combined study of electronic and phonon properties reveals that half-metallic ${\mathrm{K}}_2{\mathrm{Cr}}_8{\mathrm{O}}_{16}$, upon cooling, undergoes the correlation-assisted Peierls transition to become a Mott-Peierls ferromagnetic insulator at low temperature.

Figure 1

(a) Crystal structure of K${}_2$Cr${}_8$O${}_{16}$. Black and red lines represent the primitive and the conventional unit cell of the FM phase, respectively, while blue dotted line represents the unit cell of the FI phase. Purple, blue, sky-blue, yellow, and red balls represent K, Cr($z=0$), Cr($z=0.5$) O(1), and O(2), respectively. Right is the double chain of edge-shared CrO${}_6$ octahedra along the $c$ axis. In the FM phase, there are two types of O sites: edge-shared O(1) and corner-shared O(2). (b) The BZ of K${}_2$Cr${}_8$O${}_{16}$. Black and red lines represent the BZs of the primitive and the conventional cell of the FM phase, respectively, and the blue filled box represents the BZ of the FI phase. ($a_p^{*},b_p^{*},c_p^{*}$) and ($a_m^{*},b_m^{*},c_m^{*}$) are reciprocal lattice vectors of the primitive body-centered tetragonal unit cell of the FM phase and the monoclinic unit cell of the FI phase, respectively. Note that, in the FM phase, $\overline{M}$ of the BZ of the conventional cell is equivalent to $X$ of the BZ of the primitive cell.

Figure 2

Phonon dispersion and the corresponding partial DOS of K${}_2$Cr${}_8$O${}_{16}$ in its FM phase: (a) result from the GGA, and (b) result from the GGA + $U$ ($U=3.0$ eV). The $k$-point symmetry lines are along the BZ boundaries of the primitive unit cell of the FM phase. The negative phonon frequencies (imaginary frequencies) reflect the structural instability. It is seen from the DOS in (b) that the softened phonons are related mostly to the lattice vibrations of Cr ions.

Figure 3

(a),(b) Lattice displacements of the softened phonon mode at $X$, and (c),(d) those at $M$. Purple, blue, sky-blue, and red circles are K, Cr($z=0$), Cr($z=0.5$), and O, respectively. White circles and orange crosses at Cr and O in (a) and (c) denote the outward and inward arrows, respectively. (b) and (d) show the main atomic displacements of Cr and O(2), which lead to the formation of Cr tetramers in the four-chain column, as depicted by the gray boxes in $①$, $②$, $③$, and $④$. Thick and thin red lines represent shorter and longer bonds between Cr and O ions, respectively. Dotted gray lines connecting Cr ions in-between the double chain of (b) stand for the uniform distance between Cr-Cr, whereas thick and thin blue lines in (d) stand for the shorter and longer distances between Cr-Cr, respectively.

Figure 4

The arrangements of Cr tetramers in the ab plane obtained from the softened phonon modes. Dotted gray, thick and thin blue lines connecting Cr ions in the double chains are the same as in Fig. 3. Solid purple circles represent K atoms. (a) The stripe-type arrangement by the softened phonon at $X$. Yellow shadow indicates the modulation of the tetragonal structure in the ab plane, as observed in the experiment [2]. (b) The uniform-type arrangement by the softened phonon at $M$. (c) The checkerboard-type arrangement by the softened phonon at $\Gamma$. (d) Total DOS of each tetramer arrangement obtained in the GGA + $U$ ($U=3.0$ eV). Positive and negative DOSs represent those of spin-majority and spin-minority electrons, respectively.

Figure 5

(a) The CDD between the GGA + $U$ ($U=3.0$ eV) and the GGA for the FM phase of K${}_2$Cr${}_8$O${}_{16}$. Green and blue represent the positive and negative parts, respectively. The Cr tetramer, indicated by the red line, is seen to be connected by the $\pi$ bonding. (b) Comparison of total DOSs of the GGA and the GGA + $U$ ($U=3.0$ eV) for the FI phase of K${}_2$Cr${}_8$O${}_{16}$. The DOS of GGA-exp is obtained for the experimental structure in the GGA. (c) The GGA + $U$ ($U=3.0$ eV) band structure of the FM phase of K${}_2$Cr${}_8$O${}_{16}$, and (d) that of the FI phase of K${}_2$Cr${}_8$O${}_{16}$, both of which are plotted in the BZ of the FI phase. Band splitting is evident in the BZ boundary including $\tilde{Z}$. Spin-majority and spin-minority bands are plotted by black and red lines, respectively. The structures of FM and FI phases were optimized before the calculations except the case of GGA-exp in (b).