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We investigate the magnetic properties of Ru$_{2}$Mn$Z$ ($Z$ = Sn, Sb, Ge,Si) chemically ordered full Heusler compounds for zero as well as finitetemperatures. Based on first principles calculations we derive the interatomicisotropic bilinear and biquadratic couplings between Mn atoms from theparamagnetic state. We find frustrated isotropic couplings for all compoundsand in case of $Z$ = Si and Sb a nearest-neighbor biquadratic coupling thatfavors perpendicular alignment between the Mn spins. By using an extendedclassical Heisenberg model in combination with spin dynamics simulations weobtain the magnetic equilibrium states. From these simulations we conclude thatthe biquadratic coupling, in combination with the frustrated isotropicinteractions, leads to non-collinear magnetic ground states in the Ru$_{2}$MnSiand Ru$_{2}$MnSb compounds. In particular, for these alloys we find twodistinct, non-collinear ground states which are energetically equivalent andcan be identified as $3-q$ and $4-q$ states on a frustrated fcc lattice.Investigating the thermal stability of the non-collinear phase we find that incase of Ru$_{2}$MnSi the multiple$-q$ phase undergoes a transition to thesingle$-q$ phase, while in case of Ru$_{2}$MnSb the corresponding transition isnot obtained due to the larger magnitude of the nearest-neighbor biquadraticcoupling.

Non-collinear antiferromagnetic states in Ru-based Heusler compounds induced by biquadratic coupling