Magnetoelectric training of multiferroic domains in Mn$_2$GeO$_4$

Magnetoelectric multiferroics promise direct cross-control between coexisting ferroelectric and ferromagnetic orders, which is of interest for applications in magnetism and spintronics. A particularly interesting type of cross-control is found in spin-spiral multiferroic Mn$_2$GeO$_4$, where a ferroelectric multi-domain distribution can be globally inverted by a single magnetic field sweep. In this work we consider the initial domain evolution from zero-field cooling, imaging the evolution of domains under both magnetic and electric fields via optical second harmonic generation. We find that polarization and magnetization domains form independently when entering the multiferroic phase, and a single deterministic initialisation procedure, spanning three quarters of a field cycle, is required to achieve reliable magnetoelectric cross-coupling. This initialisation behaviour originates from a deterministic pathway from metastable to equilibrium domain patterns, in contrast to more common and less reliable domain "training" procedures that require repeated field cycles. Understanding the initial domain evolution thus enables reliable cross-control in magnetoelectric devices with highly interlinked order parameters.