Stripes in heterovalent metal-doped cuprates

Doping $La_2CuO_{4}$ with alkaline-earth, $Ae= Sr,Ba$ generates holes in $La_{2-y-x}Ln_yAe_xCuO_{4}$ ($Ln =$ $Nd,Eu$). A small fraction of the holes, $\check{p} \le$ $0.02$, suppresses 3D-AFM. The rest resides as double holes at $O$ atoms in the $CuO_2$ planes. The superlattice, formed by the $O$ atoms, gives rise to both charge order stripes and magnetization stripes with incommensurability $q_{c,m}(x) \propto \sqrt{x-\check{p}}$ for $Ae$ doping $x < \hat{x}$, but constant $q_c$ beyond. Antiparallel orientation of magnetic moments $\mathbf{m}(O)$ yields a natural explanation for the coupling of $q_m(x) = \frac{1}{2} q_c(x)$. Doping $Nd_2CuO_{4}$ with $Ce$ generates electrons that reside pairwise in copper atoms. This accounts for the different properties of $n$-doped compounds. When $n$-doped, $q_c(x) \propto \sqrt{x}$. Above a threshold temperature $T'$, electron-hole pairs are thermally generated, but then separate to reside pairwise at $Cu$ and $O$ atoms. This breaks the locking of the incommensurability of charge order and magnetization stripes, $q_m(x) \ne \frac{1}{2} q_c(x)$.