Giant orthorhombic distortions by Cu$^{2+}$ in ferrimagnetic spinel Mn$_3$O$_4$

Mn$_3$O$_4$ is a tetragonal ($c > a$) spinel that exhibits noncollinear Yafet-Kittel ferrimagnetic ordering at low temperatures. We report large orthorhombic distortions in its ferrimagnetic phase stabilized by a few percent of Cu doping. The orthorhombic strains of the ferrimagnetic phases increased linearly to the doping and reached up to $\epsilon \approx 8.2 \times 10^{-3}$ for $x$ = 0.19, which is three times larger than the saturated value under external magnetic fields. For high doping ($x \agt 0.17$), the distortions first appeared in the paramagnetic phases and underwent further enhancement simultaneously with the onset of the noncollinear ferrimagnetic ordering. We present the rich magnetostructural phase diagram of Cu$_x$Mn$_{3-x}$O$_4$, and argue that the diluted $t_2$ orbital degeneracy of Cu$^{2+}$ under tetrahedral crystal field breaks the global symmetry and triggers the orthorhombic instability inherent in Mn$_3$O$_4$.