Towards a microscopic model of magnetoelectric interactions in ${\mathrm{Ni}}_3{\mathrm{V}}_2{\mathrm{O}}_8$

We develop a microscopic magnetoelectric coupling in ${\mathrm{Ni}}_3{\mathrm{V}}_2{\mathrm{O}}_8$ (NVO) which gives rise to the trilinear phenomenological coupling used previously to explain the phase transition in which magnetic and ferroelectric order parameters appear simultaneously. Using combined neutron scattering measurements and first-principles calculations of the phonons in NVO, we identify eleven phonons which can induce the observed spontaneous polarization. A few of these phonons can actually induce a significant dipole moment. Using the calculated atomic charges, we find that the required distortion to induce the observed dipole moment is very small $(\sim 0.001\mathrm{\AA })$ and therefore it would be very difficult to observe the distortion by neutron-powder diffraction. Finally, we identify the derivatives of the exchange tensor with respect to atomic displacements, which are needed for a microscopic model of a spin-phonon coupling in NVO. We also analyze two toy models to illustrate that although the Dzyaloshinskii-Moriya interaction is often very important for coexisting of magnetic and ferroelectric order, it is not the only mechanism when the local site symmetry of the system is low enough. In fact, this coexistence can arise in NVO only due to the symmetric exchange anisotropies.

Figure 1

(Color online) Positions of the Ni(2) spine (S) and Ni(1) cross-tie (C) sites in NVO. The sublattices are numbered as in Eq. (2), below. To make contact with Tables , take the origin to be at the cross-tie site in the box. The buckling is represented by the off set $\delta =0.13b$ of the spine sites. The cross-tie sites have zero off set and are located at $y=0$, $y=b∕2$, and $y=b$.

Figure 2

Schematic diagram showing the $x$ and $y$ components of the spins in the LTI phase. We used the parameters: $q=0.28(2\pi ∕a)$, $a_{s,x}=1.6$, $a_{c,y}=1.4$, $b_{s,y}=1.3$, $b_{c,x}=-2.2$, and ${\varphi }_{\mathit{LTI}}={\varphi }_{\mathit{HTI}}+\pi ∕2$ [see Eq. (1)]. The small $z$ components of spin are not represented. The planes are buckled, so that alternately spine chains are displaced above and below the planes shown (but this buckling is not shown). In the HTI phase the cross-ties have negligible moments and the spine chains have the incommensurately modulated longitudinal moments similar to those shown. The a spin components are odd under $2_x$ and the $b$ spin components are even under $2_x$, where $2_x$ is a twofold rotation about the $x$ axis. Both spin components (which are pseudovectors) are even under ${\tilde{m}}_z$.

Figure 3

Generalized displacements $y_7$, $z_7$, $z_8$, and $x_7$ which transform like the components of a vector, for cross-tie sites (upper panels) and spine sites (lower panels). Atomic displacements (for the GD’s indicated by the labels) in the $x\text{−}z$ plane are represented by arrows, whereas those in the $+y$ direction ($-y$ direction) are indicated by crossed (open) circles.

Figure 4

As Fig. 3. Here we illustrate schematically the vector GD’s $z_9$, $z_{10}$, $z_{11}$, $y_8$, $y_9$, and $y_{10}$ for $f$ sites. The placement of the sites reproduces the symmetry of an $f$ site and is not quantitative for either V or O atoms. Three distinct $f$ sites are occupied, one by a V atom and the other two by O atoms.

Figure 5

As Fig. 4. Here we illustrate schematically the vector GD’s $y_{11}$, $x_8$, $x_9$, $z_{12}$, $y_{12}$, and $z_{13}$ for $g$ sites.

Figure 6

(Color online) The observed and calculated INS spectrum. The black vertical bars show the $B_{2u}$ phonons whose intensity is proportional to the induced dipole moment when the system is distorted by the zero-point rms values of the modes.

Figure 7

A schematic representation of the top view of two particularly interesting $B_{2u}$ modes whose displacement vectors are given in Table . The figure shows the two oxygen and two Ni atoms on $ab$ plane $(z\sim 0.25)$. The oxygen on the left is connected to V atom. The oxygen on the right is connected to cross-tie Ni which is below the oxygen atom.

Figure 8

(Color online) Diagram of an $\mathbf{a}\text{−}\mathbf{c}$ plane used to specify nearest-neighbor and next-nearest-neighbor interactions along a single spine. Circles are spine sites and square are cross-tie sites and $d=0.13b$ (see Table ). The dashed rectangle indicates the unit cell. Interactions in other $\mathbf{a}\text{−}\mathbf{c}$ planes are obtained by using translation symmetry.

Figure 9

(Color online) The two toy models. In each case the unit cell is bounded by a dashed line. The filled circles are Ni sites and the squares are O sites.

Figure 10

Generalized displacements which transform like vectors. Open circles (circles with inscribed “$x$”) are displacements out of (into) the page. The sites at positive or negative $z$ are indicated. The upper panels show modes in which the atoms move only in the $z$ direction. Upper left: an $x$-like mode $x_3$. Upper right: a $y$-like mode $y_3$. The lower panels show $z$-like modes. Left: $z_3$ with motion only along the $x$-axis and right: $z_4$ with motion only along the $y$ axis.

Figure 11

(Color online) As Fig. 8. The eight different nearest-neighbor spine-cross tie interactions are labeled $A\cdots H$. The arrow points from the first site index to the second site index. We also give the symmetry operation one has to apply to interaction $A$ to get each of the other interactions.