Identifying Landau order parameters and their transformation properties for complex multiferroics: The case of ${\mathrm{Mn}}_2{\mathrm{GeO}}_4$

For the case of ${\mathrm{Mn}}_2{\mathrm{GeO}}_4$ we characterize the complex magnetic phase that exists at temperatures below 5.5 K by order parameters for both the commensurate and the incommensurate ordering. For the incommensurate ordering we are forced to consider the transformation properties which interrelate magnetic modes at different noncollinear members of the star of the incommensurate wave vector. The known transformation properties of the underlying magnetic wave functions are used to deduce the transformation properties of the order parameters. These results are applied to construct the high-order invariants in the free energy which have been used elsewhere to describe the characteristics of switching between different domains. The properties of different domains are discussed in detail.

Figure 1

Magnetoelectric phase diagram of MGO [15]. First-order (continuous) transitions are indicated by dashed (full) lines. Phase D has no long-range magnetic or electric order. Only in phase A (below 5.5 K) does the system exhibit incommensurate magnetic order together with a spontaneous electric polarization along the $z$ axis. Phase A also exhibits commensurate magnetic order of irreps $X_1$ and $X_3$. Phases B and C (not considered in the present paper) display commensurate magnetic order.

Figure 2

Magnetic (Mn) ions in the unit cell of MGO. Filled circles (triangles or squares) represent $ch$ ($pl$) sites. For $\epsilon =0$ all the ions would be in the three planes $z=0,z=1/2$, and $z=1$. Because $\epsilon$ is positive but very small, the Mn ions represented by a triangle are displaced (by $\epsilon$) to the right of the plane and those represented by a square are displaced (by $\epsilon$) to the left of the plane.

Figure 3

Star of the wave vector ${\mathbf{k}}_A=(0.136,0.211,0)$ [15].

Figure 4

Magnetic structure of one of the two helices in the unit cell associated with wave vector ${\mathbf{k}}_A$.