Landau analysis of the symmetry of the magnetic structure and magnetoelectric interaction in multiferroics

This paper presents a detailed instruction manual for constructing the Landau expansion for magnetoelectric coupling in incommensurate ferroelectric magnets, including ${\mathrm{Ni}}_3{\mathrm{V}}_2{\mathrm{O}}_8$, $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$, $\mathrm{Mn}\mathrm{W}{\mathrm{O}}_4$, $\mathrm{Tb}{\mathrm{Mn}}_2{\mathrm{O}}_5$, $\mathrm{Y}{\mathrm{Mn}}_2{\mathrm{O}}_5$, $\mathrm{Cu}\mathrm{Fe}{\mathrm{O}}_2$, and $\mathrm{Rb}\mathrm{Fe}{\left(\mathrm{M}{\mathrm{O}}_4\right)}_2$. The first step is to describe the magnetic ordering in terms of symmetry adapted coordinates which serve as complex-valued magnetic order parameters whose transformation properties are displayed. In so doing, we use the previously proposed technique to exploit inversion symmetry, since this symmetry has seemingly been universally overlooked. Inversion symmetry severely reduces the number of fitting parameters needed to describe the spin structure, usually by fixing the relative phases of the complex fitting parameters. By introducing order parameters of known symmetry to describe the magnetic ordering, we are able to construct the trilinear magnetoelectric interaction which couples incommensurate magnetic order to the uniform polarization, and thereby we treat many of the multiferroic systems so far investigated. In most cases, the symmetry of the magnetoelectric interaction determines the direction of the magnetically induced spontaneous polarization. We use the Landau description of the magnetoelectric phase transition to discuss the qualitative behavior of various susceptibilities near the phase transition. The consequences of symmetry for optical properties such as polarization induced mixing of Raman and infrared phonons and electromagnons are analyzed. The implication of this theory for microscopic models is discussed.

Figure 1

(Color online) Inverse susceptibility ${\chi }^{-1}(q,0,0)$. (top) Ferromagnetic model $(J_1<0)$, (middle) antiferromagnetic model $(J_1<0)$, and (bottom) model with competing interactions (the nn interaction is antiferromagnetic). In each panel, one sees three groups of curves. Each group consists of the three curves for ${\chi }_{\alpha \alpha }\left(q\right)$ which depend on the component label $\alpha$ due to the anisotropy. The $x$ axis is the easiest axis and the $z$ axis is the hardest. (If the system is orthorhombic, the three axes must all be inequivalent.) The solid curves are for the highest temperature, the dashed curves are for an intermediate temperature, and the dash-dot curves are for $T=T_c$, the critical temperature for magnetic ordering. The bottom panel illustrates the nontrivial wave vector selection which occurs when one has competing interactions.

Figure 2

(Color online) Ni sites in the conventional unit cell of NVO. The primitive translation vectors ${\mathbf{v}}_n$ are ${\mathbf{v}}_1=(a∕2)\widehat{a}+(b∕2)\widehat{b}$, ${\mathbf{v}}_2=(a∕2)\widehat{a}-(b∕2)\widehat{b}$, and ${\mathbf{v}}_3=c\widehat{c}$. The “cross-tie” sites blue online c1 and c2 lie in a plane with $b=0$. The “spine” sites red online are labeled s1, s2, s3, and s4 and they may be visualized as forming chains parallel to the $\mathbf{a}$ axis. These chains are in the buckled plane with $b=\pm \delta$, where $\delta =0.13b$ as is indicated. Cross-tie sites in adjacent planes are displaced by $(\pm b∕2)\widehat{b}$. Spine sites in adjacent planes are located directly above (or below) the sites in the plane shown. In the incommensurate phases, the wave vector describing magnetic ordering lies along the $\mathbf{a}$ axis. The axis of the twofold rotation about the $x$ axis is shown. The glide plane is indicated by the mirror plane at $z=\frac{3}{4}$ and the arrow above $m_z$ indicates that a translation of $b∕2$ in the $y$ direction is involved.

Figure 3

Schematic phase diagram for NVO for a magnetic field applied along the $\mathbf{c}$ direction, taken from Ref. 6. Here, AF is an antiferromagnetic phase with a weak ferromagnetic moment, P is the paramagnetic phase, HTI is the “high-temperature incommensurate” phase in which the moments are essentially aligned along the $\mathbf{a}$ axis with a sinusoidally modulated amplitude (according to irrep ${\Gamma }_4$), and LTI is the “low-temperature incommensurate” phase in which transverse order along the $\mathbf{b}$ axis appears to make an elliptically polarized order-parameter wave (according to irreps ${\Gamma }_4$ and ${\Gamma }_1$). A spontaneous polarization $\mathbf{P}$ appears only in the LTI phase with $\mathbf{P}$ along $\mathbf{b}$.

Figure 4

(Color online) Mn sites (smaller circles, red online) and Tb sites (larger circles, blue online) in the primitive unit cell of $\mathrm{Tb}\mathrm{Mn}{\mathrm{O}}_3$. The Tb sites are in the shaded planes at $z=n\pm \frac{1}{4}$ and the Mn sites are in planes $z=n$ or $z=n+\frac{1}{2}$, where $n$ is an integer. The incommensurate wave vector is along the $\mathbf{b}$ axis. The mirror plane at $z=1∕4$ is indicated and the glide plane $m_x$ is indicated by the mirror plane at $x=3∕4$ followed by a translation (indicated by the arrow) of $b∕2$ along the $y$ axis.

Figure 5

Schematic phase diagram for TMO for magnetic fields up to $14\mathrm{T}$ applied along the $\mathbf{a}$ direction, taken from Ref. 48. Here, P is the paramagnetic phase, HTI is the “high-temperature” incommensurate phase in which (Ref. 3) the moments are essentially aligned along the $\mathbf{b}$ axis with a sinusoidally modulated amplitude according to irrep ${\Gamma }_3$, and LTI is the “low-temperature” incommensurate phase in which (Ref. 3) transverse order along the $\mathbf{c}$ axis appears to make an elliptically polarized order-parameter wave according to irreps ${\Gamma }_3$ and ${\Gamma }_2$. A spontaneous polarization $\mathbf{P}$ appears only in the LTI phase with $\mathbf{P}$ along the $\mathbf{c}$ axis for low magnetic field (Ref. 3).

Figure 6

(Color online) Two representations of $\mathrm{Tb}{\mathrm{Mn}}_2{\mathrm{O}}_5$. Top: Mn sites (red online) with smaller circles $\left({\mathrm{Mn}}^{3+}\right)$ and larger circles $\left({\mathrm{Mn}}^{4+}\right)$ and Tb sites (squares, blue online blue) in the primitive unit cell of $\mathrm{Tb}{\mathrm{Mn}}_2{\mathrm{O}}_5$. The ${\mathrm{Mn}}^{+4}$ sites are in the shaded planes at $z=n\pm \delta$ with $\delta \approx 0.25$ and the ${\mathrm{Mn}}^{+3}$ sites are in planes $z=n$, where $n$ is an integer. The Tb ions are in the planes $z=n+\frac{1}{2}$. The glide plane $m_x$ is indicated by the mirror plane at $x=3∕4$ followed by a translation (indicated by the arrow) of $b∕2$ along the $y$ axis and similarly for the glide plane $m_y$. Bottom: Perspective view. Here, the ${\mathrm{Mn}}^{3+}$ are inside oxygen pyramids of small balls and the ${\mathrm{Mn}}^{4+}$ are inside oxygen octahedra.

Figure 7

(Color online) Top: The spin structure of the ${\mathrm{Mn}}^{3+}$ ions in $\mathrm{Y}{\mathrm{Mn}}_2{\mathrm{O}}_5$ (limited to one $\mathbf{a}\text{−}\mathbf{b}$ plane), taken from Fig. 2 of Ref. 58. The sublattices are labeled in our convention. Bottom left: The spin structure after transformation by $m_x$. Bottom right: The spin structure after transformation by $m_y$.

Figure 8

(Color online) Temperature $\left(T\right)$ versus magnetic field $\left(B\right)$ phase diagram of $\mathrm{Cu}\mathrm{Fe}{\mathrm{O}}_2$ with $B$ applied along the $\mathbf{c}$ axis from Kimura et al. (Ref. 10). The upper inset shows the crystal structure of $\mathrm{Cu}\mathrm{Fe}{\mathrm{O}}_2$ and the lower insets show the magnetic structure of the commensurate states, where white and black circles correspond to the positive and negative $c$ directions. Note in the lower left inset that the hexagonal ⟨110⟩ direction (along which $q$ is oriented) is a nearest neighbor direction.

Figure 9

(Color online) The unit cell of RFMO in the $P\overline{3}$ phase. The large balls (pink online) represent the magnetic Fe ions, the small balls (blue online) represent oxygen ions, and each tetrahedron (green online) contains a Mo ion. For clarity, the Rb ion (which sits between the two tetrahedra) is not shown. The in-plane antiferromagnetic interaction $J$ is dominant. In the high-temperature $P\overline{3}m1$ phase, $J_3=J_4$, but in the presence of the lattice distortion to the $P\overline{3}$ phase, $J_3\ne J_4$ (Ref. 67). The $\mathbf{a}$ axis is parallel to the bond labeled $J$.

Figure 10

A schematic phase diagram of RFMO for magnetic fields of up to about $10\mathrm{T}$ along the $\mathbf{c}$ axis, based on Refs. 8, 67, 68, 69, 70. Here, $P$ is the paramagnetic phase and IC-TRI is an incommensurate phase described in the text in which each plane consists of the so-called 120° triangular lattice structure. CAF is a commensurate antiferromagnet phase and ICAF an incommensurate antiferromagnetic phase, neither of which is discussed in the present paper. We omit reference to subtle phase distinctions discussed in Refs. 68, 69.

Figure 11

(Color online) The 120° phase of a triangular lattice. The orientations of the spins are given by the phase $\psi \left(\mathbf{r}\right)$, defined in Eq. (157) below, for $q_{z}z+\varphi =0$. The dashed lines indicate the two-dimensional unit cell. The plus and minus signs indicate whether the oxygen tetrahedron closest to the center of the triangle is above (plus) or below (minus) the plane of the paper.

Figure 12

(Color online) Helical spin structure of RFMO. As one moves from one triangular lattice plane to the next, the spins are rotated through an angle 166° (Refs. 8, 67).

Figure 13

(Color online) The first Brillouin zone (the hexagon) and the reciprocal lattice (the dots) for a triangular lattice. The points labeled $X_1$ are all equivalent and similarly for the points labeled $X_2$. Here, $\mid {\mathbf{X}}_n\mid =4\pi ∕\left(\sqrt{3}a\right)$. The reciprocal lattice is rotated by 30° with respect to the direct lattice. In reciprocal lattice units, ${\mathbf{X}}_2=(1∕3,1∕3,0)$.

Figure 14

(Color online) After Ref. 58: the thin line is the spontaneous polarization along the $\mathbf{b}$ axis from Ref. 12 and the filled circles are from a microscopic model of Ref. 58 described in the text. The dashed line indicates the commensurate to incommensurate phase transition at about $23\mathrm{K}$.

Figure 15

(Color online) Schematic diagram of the frequency and infrared absorption cross section of a mode which is Raman active in the paraelectric phase for $T>T_F$. Note the change in slope of the frequency when the ferroelectric phase is entered. We assume the mean-field estimate for the order parameter: $P\propto {(T_F-T)}^{1∕2}$.

Figure 16

(Color online) Schematic diagram of the spin wave functions within the unit cell of NVO for the various irreps. For simplicity, only the Ni spine sites at $\mathbf{r}={\mathbf{r}}_{s,n}$ for $n=1,2,3,4$ (see Table ) are shown. The $x$ and $z$ axes are indicated and the positive $y$ axis is into the paper. (Filled circles represent spin components into the paper and crosses represent spin components out of the paper.) This figure is a pictorial representation of the data of Table . In the HTI phase, the spin distribution is that of ${\Gamma }_4$ within which the $x$ component is dominant.