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

TL;DR

This work addresses reliable magnetoelectric cross-control in a spin-spiral multiferroic by observing Mn$_2$GeO$_4$ with optical SHG after zero-field cooling. The authors demonstrate that ferroelectric $P$ and magnetization $M$ form largely independent domains upon entering the multiferroic phase and introduce a deterministic initialization spanning a full magnetic-field cycle to globally minimise the trilinear coupling term $F_{\text{ME}}\propto-\mathcal{C}\cdot M(H)\cdot P(E)$. This two-step initialization—first poling to a multi-$P$–single-$M$ state and then reversing $M$—yields a reproducible, equilibrium domain configuration that supports repeatable magnetoelectric interconversion, contrasting with conventional stochastic domain-training procedures. The work combines phenomenological and microscopic models (cone-axis dynamics and $\mathcal{C}$-dependent chirality) to explain the irreversible yet deterministic domain evolution, highlighting the importance of non-equilibrium domain evolution for reliable device functionality. The findings offer a generalizable protocol for achieving robust magnetoelectric control in complex order-parameter materials and heterostructures.

Abstract

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.

Figures (7)

• Figure 1: Optical properties of Mn$_2$GeO$_4$. (a) Linear optical absorption coefficient $\alpha(\hbar\omega)$ with light polarized along the $a$, $b$, and $c$ axes of orthorhombic Mn$_2$GeO$_4$ (linear scale on left). Central energies and width of the optical excitations are indicated by bars at the bottom (Tab. \ref{['tab:Mn2GeO4:optical_transitions']}). M1 The blue peak indicates the spectral dependence of the SHG response of $\chi_{cbb}$ at $2\hbar\omega$ (linear scale on right). (b) Temperature dependence of the SHG signal $\chi_{cbb}\propto P$ at 1.85eV. The signal is present in the multiferroic phase below $T_{\rm N}$ only, and couples to the ferroelectric polarisation $P$. (c) Crystal structure of Mn$_2$GeO$_4$, with centric Mn1 and acentric Mn2 sites in grey and black, respectively. (d,f) SHG transmission setup for domain imaging of $a$-cut and $c$-cut samples, respectively. To observe the SHG response of the $c$-cut sample, it needs to be slightly rotated around the vertical $b$ axis so that $c$-polarized contributions to the light waves can be excited. (e,g) Rotational SHG anisotropies, measured between crossed polarizers, of the (e) $a$-cut and (g) $c$-cut sample. M2 For SHG components $\chi_{cbb}$ and $\chi_{bbc}$ the incident light field $\mathbf{E}(\omega)$ is polarized parallel to the Mn2-O-Mn2 bonds (red) that promote strong antiferromagnetic exchange.
• Figure 2: Polarization domains upon magnetization reversal. Evolution of a multi-$P$--single-$M$ domain pattern obtained after an initial field cycling with (a-h) rising and (j-q) falling applied magnetic field $H_c$. Neighboring $\pm P$ domains appear with equal brightness separated by black lines. Intermediate magnetic fields lead to multi-$P$--multi-$M$ domain patterns that differ in subsequent field cycles, as demonstrated by the transient ferroelectric domains. After a complete magnetization reversal, that is, comparing panel (a) to (h) and (j) to (q), the domain pattern is largely recovered (first column), but with the polarity of each local domain inverted, i.e., $\pm P\rightarrow\mp P$.
• Figure 3: Repeatability of magnetoelectric domain inversion. Five subsequent reversals of the ferroelectric domain pattern, indicated by the interchange of blue and yellow contrast that indicate domains of opposite polarity $\pm P\parallel c$, are shown for switched magnetization $M(H_c)$. The initial domain configuration, highlighted in light blue, corresponds to the configuration shown in Fig. \ref{['fig:contrast-reversal']}(d), obtained after a magneto-electric initialization procedure from a zero-field-cooled domain configuration.
• Figure 4: Initial domain evolution. Ferroelectric domains in $a$-cut Mn$_2$GeO$_4$ after (a-e) zero-field cooling and (f-h) $E_c$-$H_c$-field cooling at 0.6MV/m and 500mT. Blue and yellow areas denote domains of opposite polarity $\pm P$. Both morphology and size of domains change upon the first field-poling steps, before repeatable magnetoelectric domain inversion is achieved. This indicates an initial non-equilibrium distribution of $P$, $M$, and antiferromagnetic $\mathcal{C}$ domains. (f-h) Similar changes upon the first magnetization reversal are observed after field-cooling in intermediate fields (highlighted in orange). (a-e) After zero-field cooling (state with white outline), intermittent domain patterns changing in morphology and typical length scales are observed. This indicates an initial non-equilibrium distribution of $P$, $M$ and antiferromagnetic $\mathcal{C}$ domains. After a full field cycle (blue path), a remanent domain state that shows fully reversible magnetoelectric inter-conversion is reached (state with black outline). (f-h) After field cooling, an initial magnetic field induces similar domain changes as in (c-e) before stabilization of the magnetoelectric inter-conversion; in this case, only half a magnetic field cycle is required (orange path). Minor variations in (g,h) can be attributed to temperature fluctuations close to $T_{\rm N}$, possibly in combination with strain.
• Figure 5: Electric-field-induced ferroelectric domain evolution. (a-c) Sequential images show the ferroelectric domain poling, starting from a multi-$P$--multi-$M$ domain pattern obtained after zero-field cooling with increasing electric field (upper right corner, in MV/m). In the region of overlap between front and back ITO electrodes (blue outline), a single-$P$--multi-$M$ domain pattern, indicated by its uniform brightness, is obtained after application of $E>14MV/m$ and retained when the field is removed (last panel). (c-e) The single-$P$--multi-$M$ domain pattern in (c), obtained after electric field poling, can be converted to a multi-$P$--single-$M$ domain pattern via the trilinear magnetoelectric effect in Eq. (\ref{['eq:trilinear']}), as demonstrated by the reappearance of ferroelectric domains upon application of a magnetic field $H_c$ in (d) and (e).