Relationship between hole density and charge-ordering wave vector in ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$

The distribution of holes in ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ is revisited with semiempirical reanalysis of the x-ray absorption data and exact diagonalization cluster calculations. Another interpretation of the XAS data leads to much larger ladder hole densities than previously suggested. These new hole densities lead to a simple interpretation of the hole Wigner crystal recently reported with $1∕3$ and $1∕5$ wave vectors along the ladder. Our interpretation is consistent with paired holes in the rung of the ladders. Exact diagonalization results for a minimal model of the doped ladders suggest that the stabilization of spin structures consisting of 4 spins in a square plaquette as a result of resonance valence bond physics suppresses the hole crystal with a $1∕4$ wave vector.

Figure 1

(a) Crystal structure of ${\mathrm{Sr}}_{14-x}{\mathrm{Ca}}_x{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ (SCCO) viewed in perspective along the $c$ direction. (b) Sketch of the structure of chains and ladders. The orientation of the O $2p$ orbitals involved in the ZR singlets are indicated. The three different oxygen sites are identified for the ladder.

Figure 2

(Color online) XAS spectra for $\mathrm{E}\Vert c$ (left panels) and $\mathrm{E}\Vert a$ (right panels) for $x=0$ [(a), (b)], $x=4$ [(c), (d)], and $x=11$ [(e), (f)]. Squares are the experimental data. The black solid lines show theoretical curves for the total fitting. The other lines are contributions from O(ch) (red dash-dotted lines), $\mathrm{O}\left(l\right)$ (green dashed lines), $\mathrm{O}\left(r\right)$ (cyan dotted lines), and the UHB (gray solid lines).

Figure 3

(Color online) (a) Spectral weight, (b) full width at half maximum, and (c) energy of O(ch) (black solid circles), $\mathrm{O}{\left(l\right)}_c$ (red solid down triangles), $\mathrm{O}{\left(l\right)}_a$ (green solid up triangles), $\mathrm{O}\left(r\right)$ (blue squares), UHB for $\mathrm{E}\Vert c$ (cyan open up triangles), and UHB for $\mathrm{E}\Vert a$ (magenta open down triangles) as function of $x$. Blue open circles are for oxygen ions in the ladders.

Figure 4

(Color online) [(a), part i to part iii] Scenarios for the ladder hole distribution in a HC with $\lambda =4c_L$. Filled (open) circles represent electrons (holes). Arrows indicate the spin of unpaired electrons. (b) The estimated number of holes in the ladders (filled circles) and the chain (open circles) per unit cell. Dashed lines show the number of holes for scenario [(a), part i], for ${\lambda }_{HC}∕c_L=3$, 4, and 5. [(c), part i] Disordered and [(c), part ii] ordered hole arrangement corresponding to ${\lambda }_{HC}=4c_L$. The disordered state has equal numbers of two-rung and four-rung spin plaquettes. (d) GS energy per rung, in units of $J_{\Vert }$, as a function of the number of rungs in the spin plaquette. Black solid circles, red open circles, green solid squares, blue open squares, cyan solid triangles, and magenta open triangles are results for $J_{\perp }∕J_{\Vert }=1.2$, 1.0, 0.8, 0.6, 0.4, and 0.2, respectively. (e) $e_{\mathit{dis}}-e_{\mathit{ord}}$ as a function of $J_{\perp }∕J_{\Vert }$, for $J_{\Vert }=130\mathrm{meV}$ (see text for details). (f) $e_{GS}\left(N_R\right)$ vs $N_R$ for $J_{\perp }∕J_{\Vert }=0.4$ and $J_{\text{ring}}∕J_{\Vert }=-0.1$, 0, and 0.1 (black solid circles, red open circles, and green solid squares, respectively).