Stripes in heterovalent metal-doped cuprates

Doping $L{a}_{2}Cu{O}_4$ with alkaline-earth, $Ae=Sr,Ba$ generates holes in $L{a}_{2-y-x}L{n}_{y}A{e}_{x}Cu{O}_4$ ($Ln=$ $Nd,Eu$). 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 $Cu{O}_2$ 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 $x < \hat{x}$,butconstant$q_c$ 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 $N{d}_{2}Cu{O}_4$ 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 $T^{\mathrm{\prime }}$, 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, $q_m(x)\mathrm{\ne }\frac{1}{2}{q}_c(x)$.