Effect of double exchange on magnetic properties of ${\mathrm{Cu}}_x{\mathrm{Zn}}_{1-x}{\mathrm{Cr}}_2{\mathrm{Se}}_4$

The results of extensive studies of magnetic properties for the spinels ${\mathrm{Cu}}_x{\mathrm{Zn}}_{1-x}{\mathrm{Cr}}_2{\mathrm{Se}}_4$ are summarized. They concern the isotherms of static magnetization in applied induction up to 14 T, the Néel temperature $T_N$ (for $x\le 0.1$) and the Curie temperature $T_C$ (for $x\ge 0.2$), and the hightemperature susceptibility, all in the full concentration range $0\le x\le 1$. We determine first the concentration ratio [${\mathrm{Cr}}^{4+}$]: [${\mathrm{Cr}}^{3+}$] from the concentration ($x$) dependence of the saturation magnetization at low temperatures and show that each Cu atom introduced to the system produces on average one ${\mathrm{Cr}}^{4+}$ ion. This means that Cu is mainly monovalent and the double exchange between ${\mathrm{Cr}}^{3+}$-${\mathrm{Cr}}^{4+}$ pairs takes place. This prediction has been tested further by calculating the ground-state spin configuration characterized by conical $\theta \mathrm{}\left(x\right)\mathrm{}$ and spiral $\Phi \mathrm{}\left(x\right)\mathrm{}$ angles, as well as the high-temperature susceptibility with both double exchange and superexchange taken into account, and by comparing them subsequently with the experimental data. The biquadratic exchange contribution was also included and proved to be essential in bringing to close numerical agreement the theory and the experimental data. From this comparison we have determined the width of the $3d$ band due to Cr ions, which is $W_d=0.63\mathrm{}\pm 0.05$ eV, and the biquadratic exchange constant $p=0.41\mathrm{}$ K. Additionally, we have determined the partial contributions to the $3d$-band width coming from the first three coordination spheres. Our analysis shows quantitatively that the double exchange depending strongly on Cu concentration drives the transition from helimagnetism at $x=0\mathrm{}$ through a conical structure to ferromagnetism at $x\ge 0.8$.