Magnetic order, hysteresis, and phase coexistence in magnetoelectric ${\mathrm{LiCoPO}}_4$

The magnetic phase diagram of magnetoelectric ${\mathrm{LiCoPO}}_4$ is established using neutron diffraction and magnetometry in fields up to $25.9\mathrm{T}$ applied along the crystallographic $b$ axis. For fields greater than $11.9\mathrm{T}$, the magnetic unit cell triples in size with propagation vector $\mathbf{Q}=(0,\frac{1}{3},0)$. A magnetized elliptic cycloid is formed with spins in the $(b,c)$ plane and the major axis oriented along $b$. Such a structure allows for the magnetoelectric effect with an electric polarization along $c$ induced by magnetic fields applied along $b$. Intriguingly, additional ordering vectors $\mathbf{Q}\approx (0,\frac{1}{4},0)$ and $\mathbf{Q}\approx (0,\frac{1}{2},0)$ appear for increasing fields in the hysteresis region below the transition field. Traces of this behavior are also observed in the magnetization. A simple model based on a mean-field approach is proposed to explain these additional ordering vectors. In the field interval $20.5–21.0\mathrm{T}$, the propagation vector $\mathbf{Q}=(0,\frac{1}{3},0)$ remains but the spins orient differently compared to the cycloid phase. Above $21.0\mathrm{T}$ and up until saturation, a commensurate magnetic structure exists with a ferromagnetic component along $b$ and an antiferromagnetic component along $c$.

Figure 1

Magnetic phase diagram of ${\mathrm{LiCoPO}}_4$ for fields up to $25.9\mathrm{T}$ applied along $b$ as measured by magnetization (square symbols) and neutron diffraction (triangular symbols). The gray symbols are from Ref. [8]. The transition fields and temperatures are determined for increasing fields and upon cooling below $16\mathrm{T}$ and upon heating above. The propagation vectors identified from neutron diffraction are shown in the respective phases. The zero-field magnetic unit cell with exchange interactions indicated is shown as an inset. Note that only the major spin component along $b$ is shown (see text).

Figure 2

Magnetization and neutron diffraction data from RITA-II as a function of field for both increasing (green squares) and decreasing (blue triangles) field. (a), (b) Show the magnetization at 3 and $6\mathrm{K}$, respectively. The material exhibits hysteresis and the inset and arrows emphasize the special features at $3\mathrm{K}$ discussed in Sec. 3. These features are absent at $6\mathrm{K}$. The dashed lines indicate $\frac{1}{3}$ and $\frac{1}{4}$ of the saturation magnetization. (c), (d) Show the integrated intensity of the magnetic Bragg peak $(3,0,1)$ at 1.5 and $6\mathrm{K}$, respectively. The solid lines are guides to the eye. Blue and green arrows indicate the field ramp directions.

Figure 3

Neutron scattering intensity as a function of $(3,K,1)$ for selected fields at $2\mathrm{K}$ for (a)–(c) increasing and (d)–(f) decreasing field. The selected fields are in the low-field commensurate phase (top panels), in the transition region (middle panels), and in the $\mathbf{Q}=(0,\frac{1}{3},0)$ phase (bottom panels). The actual field values are given in the plots. The solid lines are fits to the respective data sets as described in the text.

Figure 4

Color plots of the intensity of $(3,K,1)$ as a function of magnetic field applied along $b$ at (a), (b) $2\mathrm{K}$ and (c), (d) $6\mathrm{K}$ for both increasing and decreasing field as measured at D23. The white crosses to the right in each color plot denote the field values for which scans have been performed. Note the relatively few points in (d) and the difference in maximum field between the top and bottom panels. No data were collected in the hatched area.

Figure 5

Refined magnetic structure. (a) Calculated vs observed scattering intensities for the collected incommensurate peaks as obtained in fullprof for the refined magnetic structure. The dashed line shows $|F_{\mathrm{calc}}{|}^2={\left|F_{\mathrm{obs}}\right|}^2$. (b), (c) Magnetic structure for $11.9–20.5\mathrm{T}$ applied along $b$ shown in the $(b,c)$ plane and in 3D, respectively. The spins order in a superposition of an elliptic cycloid and a ferromagnetic component along $b$. This results in $\frac{2}{3}$ of the spins being almost parallel and $\frac{1}{3}$ being antiparallel to the field direction. In (b) the nearest- and next-nearest-neighbor interactions $J_{bc}$ and $J_b$ are shown.

Figure 6

Neutron diffraction results from HFM/EXED. (a) Neutron counts as a function of $(0,K,0)$ around $K=1$ (left panels) and $K=\frac{4}{3}$ (right panels) at selected field values. Linear backgrounds have been fitted and subtracted for each data set. The orange lines are subsequential Gaussian fits. (b), (c) Integrated intensity as a function of magnetic field up to $25.9\mathrm{T}$ of $(0,1,0)$ and $(0,\frac{4}{3},0)$, respectively. The star symbol in (b) shows the expected zero intensity of $(0,1,0)$ at saturation [28]. The dashed line is a guide to the eye.

Figure 7

Stacking faults. (a) Possible stacking fault structures with period $n\in \mathbb{N}$ for $n>1$ and with magnetization $(1/n)M_S$. Spin direction is denoted by the ion color: green (along $b)$ and blue (along $\overline{b})$. Only one layer of ions in the $(b,c)$ plane is shown here. (b) Energy per magnetic ion as a function of applied field calculated from the stacking fault structures. $J_{bc}=2J_b$ is assumed. The zero-field energy $E_1$ is shown with the solid blue line and $E_n\to E_1$ for $n\to \infty$. The energies for $n>2$ cross at this level exactly at the transition field $H_C=11.9\mathrm{T}$ as shown with the vertical line.