Figure 1

(a) A polymer chain in the standard orientation. The carbon atoms involved in the cyloaddition reactions form rectangles (gray) lying in a same plane — the cycloaddition plane — which, for the standard orientation, is parallel to the $(\vec{a},\vec{c})$ crystal plane. Projections of the crystal structure of polymerized (b) $\mathrm{K}{\mathrm{C}}_{60}$ and (c) Rb- and $\mathrm{Cs}{\mathrm{C}}_{60}$ onto the $(\vec{b},\vec{c})$ plane. The cycloaddition planes, perpendicular to the $(\vec{b},\vec{c})$ plane, are marked by thick bars. The bigger dots represent the alkali-metal atoms. The ${\mathrm{C}}_{60}$ monomers and the alkali-metal atoms shown in black display a shift of $\vec{a}∕2$ along the $\vec{a}$ direction with respect to those shown in gray.Reuse & Permissions

Figure 2

Superstructure formation in polymerized $\mathrm{K}{\mathrm{C}}_{60}$ at $T_c\approx 60\mathrm{K}$. Above $T_c,$ $\vec{a},\vec{b}$, and $\vec{c}$ are the primitive basis vectors; below $T_c$, the primitive basis vectors are $\vec{a}+\vec{c},\vec{b}$, and $\vec{a}-\vec{c}$. Polymer chains and alkali atoms in black display a shift of $\vec{b}∕2$ with respect to those in gray.Reuse & Permissions

Figure 3

${\left({\mathrm{C}}_{60}^{-}\right)}_n$ polymer chain with its ${\mathrm{C}}_{60}^{-}$ valence charge distributions. The charge distributions are represented by their angular parts, taken from Ref. 14, shown as contourplots on spheres centered at the ${\mathrm{C}}_{60}^{-}$ monomer lattice sites (the higher the valence electronic density, the darker the grayscale level). (a) All charge distributions are in their original orientations, the periodicity along the axis of polymerization (parallel to the $\vec{a}$ direction) is $a$. (b) An OCDW — consisting of alternating rotational deviations $\pm \Delta {\psi }_0$ of the charge distributions with the polymer chain axis (dashed line) as rotation axis — the periodicity along $\vec{a}$ doubles. Only the charge distributions rotate; the ${\left({\mathrm{C}}_{60}\right)}_n$ core structure is considered to be fixed.Reuse & Permissions

Figure 4

Peierls instability. A linear chain of originally equidistant atoms displays a longitudinal displacement wave with wave vector $k=2\mid k_F\mid$ (inset). The primitive unit cell is doubled, the first Brillouin zone is reduced to half its original size, and the lowest band gets completely filled because there are two atoms of valence one per unit cell. (The original band structure, shown by thin black lines, consists of parabolas. The modified band structure is shown by thick gray lines.) The band gap $\delta$ at the zone boundary separates the exactly filled lowest band from an empty band, the system becomes insulating. (Band filling is shown in dark gray.)Reuse & Permissions

Figure 5

Band structure $\{\lambda \left(\vec{k}\right)=E\left(\vec{k}\right)-E_{t_{1u}}\}∕\mid U_0\mid C^2$ of a $\mathrm{K}{\mathrm{C}}_{60}$ crystal, taking into account interactions between polymer chains, (a) when the OCDWs are “switched off” $(\Delta {\psi }_0=0°)$ and (b) “switched on” $(\Delta {\psi }_0=13°)$. The $\vec{k}$ path for which $\left\{\lambda \left(\vec{k}\right)\right\}$ has been calculated is constructed from connecting the points $\Gamma ,Z,Y$, etc. (see Table ) of the first Brillouin zone by straight lines; the numerical calculations have been done for 500 points in every connection. For easy understanding of the correspondence between the global plots (a) and (b) and their zoom-ins $\left({\mathrm{a}}_1\right)\text{–}\left({\mathrm{a}}_6\right)$ and $\left({\mathrm{b}}_1\right)\text{–}\left({\mathrm{b}}_6\right)$, the path $\Gamma \to Z\to Y\to \cdots$ has been mapped linearly onto $0\to 1\to 2\to \cdots$ . The plots have been shifted vertically by a constant to make the (apparent) crossover of the second and third bands lie around zero. The plotted results are for $U_1∕U_0=0.9,U_2∕U_0=0.8$, and $\epsilon ={10}^{-3}$.Reuse & Permissions

Figure 6

Results of the band structure calculations (for $\epsilon ={10}^{-3}$) with $U_1∕U_0=100$ and $U_2∕U_0=1$, (a) when no OCDWs are present $(\Delta {\psi }_0=0°)$ and (b) when the OCDWs are “switched on” $(\Delta {\psi }_0=13°)$. A vertical shift has been introduced to centralize the band gap of (b).Reuse & Permissions

Figure 7

Band structure $\{\lambda \left(k\right)=E\left(k\right)-E_s\}∕\mid U_0\mid A^2$ for a linear chain of atoms as a result of the tight-binding calculations described in Sec. 3. For equally separated atoms ($\Delta =0$, dashed lines), there is no band gap at $k_F=\pm \pi ∕2a$, but when a lattice deformation with wave vector $k=2\mid k_F\mid$ is applied ($\Delta =a∕10\ne 0$, solid lines), the two bands split at the zone boundary. (a) shows the results for the most simple approximation [$\epsilon =1$, see Eqs. (3.4a, 3.4b)], while (b) shows a higher-order approximation $(\epsilon ={10}^{-3})$. The curves of the two cases are different, but from a physical point of view, the band structures are equivalent.Reuse & Permissions

Figure 8

Band structure $\{\lambda \left(\vec{k}\right)=E\left(\vec{k}\right)-E_{t_{1u}}\}∕\mid U_0\mid C^2$, when modification (4.1) of the effective potential $U\left(\vec{r}\right)$ due to the presence of the alkali ions is taken into account. (a) For $\Delta {\psi }_0=0°,\delta =0$, there is no band gap. (b) For $\Delta {\psi }_0=13°,\delta =0.25$, a gap, of reasonable magnitude compared to the overall band width, is present. Note that the absence of a band gap (a) is not due to crossover, as in Figs. 5a, 6a, but due to “vertical” overlap.Reuse & Permissions