Search for magnetoelectric monopole response in Cr$_2$O$_3$ powder

Powder samples have been suggested as a pathway to fabricate isotropic magnetoelectric (ME) materials which effectively only have a pseudoscalar or monopole ME response. We demonstrate that random distribution of ME grains alone does not warrant isotropic ME response because the activation of a non-vanishing ME response requires a ME field cooling protocol which tends to induce preferred axes. We investigate the evolution of ME susceptibility in powder chromia samples for various ME field cooling protocols both theoretically and experimentally. In particular, we work out the theoretical expressions for ME susceptibility for powder Chromia in the framework of statistical mechanics where Boltzmann factors weigh the orientation of the N\'eel vector relative to the local orientation of the c-axis of a grain. Previous approximations oversimplified the thermodynamic nature of the annealing process giving rise to misleading conclusions on the role of the magnitude of the applied product of electric and magnetic fields on the ME response. In accordance with our refined theory, a strong dependence of the functional form of $\alpha$ vs. $T$ of Chromia powders on the ME field cooling protocol is observed. It shows that Chromia powder is not generically an isotropic ME effective medium but provides a pathway to realize the elusive isotropic ME response.