Difference in magnetic and ferroelectric properties between rhombohedral and hexagonal polytypes of ${\mathrm{AgFeO}}_2$: A single-crystal study

We have investigated magnetic and dielectric properties of rhombohedral $3R-{\mathrm{AgFeO}}_2$ and hexagonal $2\mathrm{H}-{\mathrm{AgFeO}}_2$ by using magnetic and dielectric bulk measurements and a neutron diffraction experiment with single crystals grown by hydrothermal synthesis. Magnetic phase transitions occur at $T=14.0$ K and $T=6.0$ K in $3R-{\mathrm{AgFeO}}_2$ and $T=17.0$ K and $T=9.5$ K in $2H-{\mathrm{AgFeO}}_2$ under zero magnetic field. Multistep metamagnetic phase transitions were observed in $3R-{\mathrm{AgFeO}}_2$ in magnetization measurements up to 60 T, while a single phase transition occurs in $2H-{\mathrm{AgFeO}}_2$. The ferroelectric polarization parallel and perpendicular to the triangular lattice plane appears below $T=6.0\mathrm{K}$ in $3R-{\mathrm{AgFeO}}_2$, which is concomitant with the onset of the cycloid magnetic ordering with the propagation vector $\mathit{k}=(-\frac{1}{2},q,\frac{1}{2};q\simeq 0.2)$ and the magnetic point group polar $m{1}^{\prime }$. On the other hand, the ferroelectric polarization is absent even below the lower phase transition temperature in $2H-{\mathrm{AgFeO}}_2$, which can be explained by the proper screw magnetic structure with $\mathit{k}=(0,q,0;q\simeq 0.4)$ and the nonpolar ${2221}^{\prime }$ point group. Although the two-dimensional triangular lattice layers of ${\mathrm{Fe}}^{3+}$ are common in the two polytypes, the magnetic and ferroelectric properties are significantly different. The emergence of ferroelectric polarization which is not confined to be within the plane of the cycloid for $3R-{\mathrm{AgFeO}}_2$ can be explained by the extended inverse Dzyaloshinskii-Moriya effect with two orthogonal components, ${\mathit{p}}_1\propto {\mathit{r}}_{ij}\times [{\mathit{S}}_i\times {\mathit{S}}_j]$ and ${\mathit{p}}_2\propto {\mathit{S}}_i\times {\mathit{S}}_j$. Unlike other delafossite compounds, the ${\mathit{p}}_2$ component is not allowed in the proper screw phase of $2H-{\mathrm{AgFeO}}_2$ due to the symmetry restriction of the parent space group.

Figure 1

Crystal structures of (a) $3R-{\mathrm{AgFeO}}_2$ ($R\overline{3}m$) and (b) $2H-{\mathrm{AgFeO}}_2$ ($P{6}_3/mmc$). The lattice parameters for $3R-{\mathrm{AgFeO}}_2$ are $a=3.0391\AA , c=18.590\AA$, Ag(0,0,0), Fe(0,0,0.5), O(0,0,0.1112) [25], and those for $2H-{\mathrm{AgFeO}}_2$ are $a=3.039\AA , c=12.395\AA$, Ag($1/3,2/3,0.25$), Fe(0,0,0), O($1/3,2/3,0.0833$) [26].

Figure 2

Photographs and x-ray Laue images of the hexagonal $c$ direction of the single crystals of (a) $3R-{\mathrm{AgFeO}}_2$ and (b) $2H-{\mathrm{AgFeO}}_2$. Relationship of the unit cell basis vectors above and below the magnetic phase transition temperatures: (c) $R\overline{3}m$ and $C2/m$ in $3R-{\mathrm{AgFeO}}_2$ and (d) $P{6}_3/mmc$ and $Ccmm$ in $2H-{\mathrm{AgFeO}}_2$ space groups. The subscripts r, m, h, and o denote unit vectors for $R\overline{3}m$ (hexagonal setting), $C2/m, P{6}_3/mmc$, and $Ccmm$, respectively.

Figure 3

Temperature dependence of magnetic susceptibility under magnetic fields (a) perpendicular to the hexagonal $c$ axis and (b) parallel to the $c$ axis in $3R-{\mathrm{AgFeO}}_2$. (c) and (d) The data for $2H-{\mathrm{AgFeO}}_2$. These data were normalized to the value at $T=20$ K. The dotted lines show the magnetic phase transition temperatures.

Figure 4

Magnetization processes and the derivative of magnetization with respect to the magnetic field ($dM/dB$) under pulsed magnetic field (a) perpendicular to the hexagonal $c$ axis and (b) parallel to the $c$ axis at $T=1.3$ K in $3R-{\mathrm{AgFeO}}_2$. (c) and (d) The data for $2H-{\mathrm{AgFeO}}_2$. The inset in (b) shows the data measured with a powder sample of $3R-{\mathrm{AgFeO}}_2$. Triangles show the phase transition fields.

Figure 5

Temperature dependence of dielectric permittivity (a) perpendicular to the hexagonal $c$ axis and (b) parallel to the $c$ axis in $3R-{\mathrm{AgFeO}}_2$. (c) Temperature dependence of electric polarization perpendicular to the hexagonal $c$ axis and parallel to the $c$ axis, which was measured after poling in electric field, 267 and 800 kV/m, respectively. The inset in (c) is the poling electric field dependence of electric polarization along both directions. (d) Magnetic field dependence of the electric polarization along the $c$ axis at $T=4.2$ K.

Figure 6

Temperature dependence of the neutron diffraction profile along the reciprocal lattice lines (a) $[\overline{1},K,\frac{1}{2}]$ and (b) [$\frac{1}{2},q,\overline{\frac{1}{2}}]$ in $3R-{\mathrm{AgFeO}}_2$. (c) Schematic illustrations of the reciprocal lattice planes, ($2\overline{H},K,H$) for the ICM1 phase and ($\overline{H},K,H$) for the ICM2 phase. Squares and circles denote nuclear and magnetic reflections points.

Figure 7

(a) Temperature dependence of the integrated intensity of magnetic reflections indexed by $\mathit{k}=(\overline{1},q,\frac{1}{2})$ and $\mathit{k}=(\overline{\frac{1}{2}},q,\frac{1}{2})$ in $3R-{\mathrm{AgFeO}}_2$. (b) Temperature dependence of the incommensurate wave number ($b$ component) in the $\mathit{k}$ vector.

Figure 8

Temperature dependence of the neutron diffraction profile along the reciprocal lattice lines (a) $[0,K,\overline{1}]$ and (b) $[0,q,\overline{1}]$ in $2H-{\mathrm{AgFeO}}_2$. (c) Schematic illustrations of the reciprocal lattice planes, ($0,K,L$) for the ICM1 and ICM2 phases. Squares and circles denote nuclear and magnetic reflections points.

Figure 9

(a) Temperature dependence of the integrated intensity of magnetic reflections indexed by $\mathit{k}=(0,q,0)$ in $2H-{\mathrm{AgFeO}}_2$. (b) Temperature dependence of the incommensurate wave number ($b$ component) in the $\mathit{k}$ vector. Solid and open symbols denote the data measured with warming and cooling processes, respectively.

Figure 10

(a) Result of the refinement for the data acquired at $T=1.5$ K in $2H-{\mathrm{AgFeO}}_2$. The open and solid symbols denote magnetic and nuclear reflections, and the difference in the symbols for magnetic data corresponds to different domains. The refined magnetic structure parameters, magnetic domain population, magnetic moments $M_c$ and $M_a$, initial phase shift $\delta$, and reliability factor $R_F$, are listed in the inset. (b) Illustrations of the determined magnetic structure, proper screw structure with ellipsoidal distortion for the ICM2 phase in $2H-{\mathrm{AgFeO}}_2$.

Figure 11

Schematic illustrations representing the relationships between noncollinear spin modulation along the $b$ axis and the electric polarization directions determined by the extended inverse DM mechanism [19], ${\mathit{p}}_1\propto {\mathit{r}}_{ij}\times [{\mathit{S}}_i\times {\mathit{S}}_j]$ and ${\mathit{p}}_2\propto {\mathit{S}}_i\times {\mathit{S}}_j$, for (a) the cycloid phase in $3R-{\mathrm{AgFeO}}_2$ and (b) the proper screw phase in $2H-{\mathrm{AgFeO}}_2$.