Non-collinear 2k antiferromagnetism in the Zintl semiconductor Eu$_5$In$_2$Sb$_6

Eu$_5$In$_2$Sb$_6$ is an orthorhombic non-symmorphic small band gap semiconductor with three distinct Eu$^{2+}$ sites and two low-temperature magnetic phase transitions. The material displays one of the greatest (negative) magnetoresistances of known stoichiometric antiferromagnets and belongs to a family of Zintl materials that may host an axion insulator. Using single crystal neutron diffraction, we show that the $T_{\mathrm{N1}}=14\mathrm{~K}$ second-order phase transition is associated with long-range antiferromagnetic order within the chemical unit cell $\left( k_1 = (000) \right)$. Upon cooling below $T_{\mathrm{N1}}$, the relative sublattice magnetizations of this structure vary until a second-order phase transition at $T_{\mathrm{N2}}=7\mathrm{~K}$ that doubles the unit cell along the $\hat{c}$ axis $\left( k_2 = \left(00\frac{1}{2}\right) \right)$. We show the anisotropic susceptibility and our magnetic neutron diffraction data are consistent with magnetic structures described by the $\Gamma_3$ irreducible representation with the staggered magnetization of the $k_1$ and $k_2$ components polarized along the $\hat{b}$ and $\hat{a}$ axis, respectively. As the $k_2$ component develops, the amplitude of the $k_1$ component is reduced, which indicates a 2k non-collinear magnetic structure. Density functional theory is used to calculate the energies of these magnetic structures and to show the $k_1$ phase is a metal so $T_{\mathrm{N1}}$ is a rare example of a unit-cell-preserving second-order phase transition from a paramagnetic semiconductor to an antiferromagnetic metal. DFT indicates the transition at $T_{\mathrm{N2}}$ to a doubled unit cell reduces the carrier density of the metal, which is consistent with resistivity data.