Field-induced magnetic phases and electric polarization in ${\text{LiNiPO}}_4$

Neutron diffraction is used to probe the $\left(H,T\right)$ phase diagram of magnetoelectric (ME) ${\text{LiNiPO}}_4$ for magnetic fields along the $c$ axis. At zero field the Ni spins order in two antiferromagnetic phases. One has commensurate (C) structures and general ordering vectors ${\mathbf{k}}_{\text{C}}=\left(0,0,0\right)$; the other one is incommensurate (IC) with ${\mathbf{k}}_{\text{IC}}=\left(0,q,0\right)$. At low temperatures the C order collapses above ${\mu }_{0}H=12\text{T}$ and adopts an IC structure with modulation vector parallel to ${\mathbf{k}}_{\text{IC}}$. We show that C order is required for the ME effect and establish how electric polarization results from a field-induced reduction in the total magnetoelastic energy.

Figure 1

(Color online) (a) ${\text{Ni}}^{2+}$-spin configuration of the zero-field C structure. Ni positions are labeled $i$ according to their positions ${\mathbf{r}}_{\mathbf{i}}$ $\left(i=1,\dots ,4\right)$ but shifted $\left(-0.25,-0.25,0\right)$ compared to the values given in the text. [(b) and (c)] Projected C structure at zero and finite field $\mathbf{H}\parallel \mathbf{c}$ seen along the $b$ axis. Spin angles are exaggerated for clarity. The applied field causes the ${\text{Ni}}^{2+}$ moments to rotate, resulting in a magnetization $\mathbf{M}$ of the crystal, as described in the text. (d) Linearly polarized (LP) IC magnetic structure at zero field seen along the $a$ axis.

Figure 2

(Color online) (a) $\left(H,T\right)$ phase diagram of ${\text{LiNiPO}}_4$ for $\mathbf{H}\parallel \mathbf{c}$ including a C and an IC antiferromagnetic phase, and a paramagnetic (P) phase, which at zero field supports short-range fluctuations up to $T=40\text{K}$ (Ref. 13). [(b) and (c)] $T$ and $H$ dependences of the (0,1,0) magnetic Bragg peak intensity at three temperatures and fields.

Figure 3

(Color online) [(a) and (b)] Position and integrated intensity of the $\left(0,1+q,0\right)$ IC peak as function of temperature for different fields $\mathbf{H}\parallel \mathbf{c}$. [(c) and (d)] Scattering intensity for wave vectors $\left(0,k,0\right)$ at ${\mu }_{0}H=13.5$ and 14.7 T. At 13.5 T, the IC peak disappears near $T=15\text{K}$ where the system enters the C phase [cf. Fig. 2a]. [(e) and (f)] Background-subtracted peak intensities of the (1,1,0) peak as function of field and temperature.

Figure 4

(Color online) (a) Positions of Ni [large blue (dark) circles], O [small yellow (bright) circles], and the ${\text{PO}}_4$ tetrahedra (triangles) in zero field. (b) Same as (a), but for $\mathbf{H}\parallel \mathbf{c}$. ${\text{PO}}_4$ tetrahedra are assumed to shift downward with $x$ (arrows), giving an electric polarization $P_x$ and changing the SE interaction as explained in the text. (c) Measured (circles) (Ref. 13) and calculated ME coefficients ${\alpha }_{xz}$ for $\mathbf{H}\parallel \mathbf{c}$, assuming identical spin lengths (dashed line) and different spin lengths (solid line). (d) Same as (c) for ${\alpha }_{zx}$ with $\mathbf{H}\parallel \mathbf{a}$. Insets of (c) and (d) show the assumed angles between ${\mathbf{S}}_1$, ${\mathbf{S}}_2$, ${\mathbf{S}}_3$, and ${\mathbf{S}}_4$.