Magnetic and ferroelectric orderings in multiferroic $\alpha$-${\mathrm{NaFeO}}_2$

The triangular based antiferromagnet $\alpha$-${\mathrm{NaFeO}}_2$ has been studied by magnetization, dielectric, and neutron diffraction measurements as a function of temperature and magnetic field. The appropriate $(H-T)$ phase diagram was constructed revealing a complex behavior due to a competition between several magnetic phases. In zero field, the system undergoes a sequence of magnetostructural transitions; initially from paramagnetic $R\overline{3}m{1}^{\prime }$ phase to the incommensurate spin density wave (ICM1) at $T_{\mathrm{N}1}=11$ K with the nonpolar (3+1) magnetic superspace group $C2/m{1}^{\prime }(0,\beta ,\frac{1}{2})s0s$, then, below $T_{\mathrm{N}2}=7.5$ K, the ICM1 phase coexists with the polar cycloidal ordering (ICM2) possessing the $Cm{1}^{\prime }(0,\beta ,\frac{1}{2})0s$ superspace symmetry and finally the commensurate collinear ordering (CM) with the nonpolar magnetic space group $P_a{2}_1/m$ develops below $T_{\mathrm{N}3}=5.5$ K as the ground state of the system. A small amount of ICM2 coexists with the ICM1 and CM phases resulting in a nonzero measured polarization below $T_{\mathrm{N}2}$. Magnetic field destabilizes the collinear ground state and promotes the polar ICM2 phase resulting in a drastic increase of the polarization. The symmetry of the zero field cycloidal structure allows the two orthogonal components ${\mathit{p}}_1\propto {\mathit{r}}_{ij}\times ({\mathit{S}}_i\times {\mathit{S}}_j)$ and ${\mathit{p}}_2\propto {\mathit{S}}_i\times {\mathit{S}}_j$ to contribute to the macroscopic polarization through the inverse DM effect. The applied magnetic field reduces the symmetry of the ICM2 phase down to the triclinic $P1(\alpha ,\beta ,\gamma )0$, resulting in admixture of another cycloidal and helical components both generating magnetic field switchable polarization ${\mathit{p}}_3$ perpendicular to ${\mathit{p}}_1$ and ${\mathit{p}}_2$.

Figure 1

(Left) Ordered rock-salt type crystal structure of $\alpha$-${\mathrm{NaFeO}}_2$. (Right) Relationship between the hexagonal and monoclinic basis vectors [$\mathit{a}={\mathit{b}}_h-{\mathit{a}}_h$, $\mathit{b}={\mathit{a}}_h+{\mathit{b}}_h$, $\mathit{c}=\frac{1}{3}({\mathit{a}}_h-{\mathit{b}}_h-{\mathit{c}}_h)$, Fe is at the origin].

Figure 2

(a) Temperature dependencies of magnetization for $\alpha$-${\mathrm{NaFeO}}_2$, measured in different magnetic fields. The data were normalized to the field and scaled to match the value at $T=20$ K and $H=0.1$ T. (b) The magnetization (open circles) and its derivative with respect to the magnetic field (closed circles) at $T=2$ K. Vertical arrows indicate the temperatures and field where anomalies appear.

Figure 3

Temperature dependence of (a) electric polarization, (b) real, and (c) imaginary parts of the relative dielectric constants in magnetic fields up to 9 T for $\alpha$-${\mathrm{NaFeO}}_2$. Polarization as a function of the poling electric field $E_p$ at $H=0$ and 9 T are shown in the inset of (a).

Figure 4

Temperature dependence of the neutron powder diffraction profiles measured in zero magnetic field for $\alpha$-${\mathrm{NaFeO}}_2$. The patterns were collected using (a) 90 deg and (b) backscattering detector banks on WISH. The data at 16 K in the paramagnetic phase were subtracted from the low temperature patterns. All data were taken on heating after cooling down to the base temperature 1.5 K.

Figure 5

Schematic drawings of the reciprocal lattice zone for (a) ICM1, (b) CM, and (c) ICM2 phases. Circle and square symbols are the positions of the magnetic and nuclear reflections, respectively. Crosses in (c) indicate the projected positions of the magnetic reflection of ICM2 phase to the monoclinic ($0,K,L$) zone. Solid and dotted lines denote the monoclinic and hexagonal reciprocal lattice unit cells, respectively. The subscripts ($m/h$) and superscripts ($+/-$) indicate monoclinic/hexagonal and $+\mathit{k}/-\mathit{k}$ satellites, respectively

Figure 6

Temperature dependencies of (a) the integrated intensity of the incommensurate 110${}^{-}$ and the commensurate $-\frac{1}{2},0,\frac{1}{2}$ reflections, (b) the incommensurate wave number $q_b$, and (c) the ferroelectric polarization. Vertical dotted lines indicate the phase transition temperatures.

Figure 7

Refinement quality of the WISH data measured at (a) 9 K and 0 T and (b) 1.5 K and 0 T. The first line of the vertical bars indicate the peak positions for the crystallographic main phase of $\alpha$-${\mathrm{NaFeO}}_2$. The second line in (a), and the second and the third lines in (b) are for the spin-density wave, the collinear commensurate, and a cycloid magnetic structures, respectively. The other three lines correspond to impurity peaks of $\beta$-${\mathrm{NaFeO}}_2$, its magnetic phase, and ${\mathrm{Na}}_2$${\mathrm{CO}}_3$. The illustrations of the magnetic structures and refined parameters determined for each phase are shown in the insets.

Figure 8

Diffraction patterns for $\alpha$-${\mathrm{NaFeO}}_2$ measured in different magnetic fields using (a) 90 deg and (b) backscattering WISH detector banks, at 1.5 K. The data at 16 K in the paramagnetic phase were subtracted from the low temperature patterns. All data were measured with increasing magnetic field after cooling in zero field.

Figure 9

Magnetic field dependence of (a) the integrated intensity of the magnetic reflections indexed by the commensurate ${\mathit{k}}_{\mathrm{CM}}=(-\frac{1}{2},0,\frac{1}{2})$ and the incommensurate ${\mathit{k}}_{\mathrm{ICM}2}=({\delta }_a,q_b,\frac{1}{2}+{\delta }_c)$ propagation vectors and (b) the wave number $q_b$ at 1.5 K. (c) Magnetic field variation of the ferroelectric polarization at 2.0 K.

Figure 10

(a) Magnetic field dependence of the satellite reflections $000+{\mathit{k}}_{\mathrm{ICM}2}$ and $001+{\mathit{k}}_{\mathrm{ICM}2}$ [${\mathit{k}}_{\mathrm{ICM}2}=({\delta }_a,q_b,\frac{1}{2}+{\delta }_c)$] at 1.5 K. The intensity of the former reflection is multiplied by a constant for appropriate scaling. (b) The modified cycloid magnetic structure as the most probable model for the ICM2 phase. The magnetic moments are rotating in the $ab$ plane. (c) The magnetic structural refinement of the experimental data collected at $T=1.5$ K and $H=5$ T. The first line of the vertical bars indicates the peak positions of the main structural phase of $\alpha$-${\mathrm{NaFeO}}_2$. The second, third, and fourth bar lines correspond to the CM collinear state, $ab$ cycloid, and ferromagnetic components, respectively. The other bar lines indicate the impurity peak positions, which are explained in Fig. 7. The refined parameters and reliability factors are given in the inset.

Figure 11

Temperature dependence of the neutron powder diffraction profiles at 9 T in $\alpha$-${\mathrm{NaFeO}}_2$. The data were measured using (a) 90 deg and (b) backscattering detector banks on WISH. The data at 16 K in the paramagnetic phase were subtracted from each low temperature pattern. All data were taken on a heating process after cooling down to 1.5 K in $H=9$ T.

Figure 12

Temperature dependence of (a) the integrated intensity of the magnetic reflection 000${}^{+}$, (b) the wave number $q_b$, and the incommensurabilities ${\delta }_a$ and ${\delta }_b$, and (c) the electric polarization in $H=9$ T for $\alpha$-${\mathrm{NaFeO}}_2$.

Figure 13

Magnetic phase diagram of polycrystalline sample of $\alpha$-${\mathrm{NaFeO}}_2$. Circle, triangle, and square symbols denote the temperatures and fields where the magnetization, dielectric constant, and neutron diffraction data show anomaly, respectively. The phase boundaries are drawn by dotted lines. In the regions where the phase coexistences were observed, the minor phase is given in parentheses. The hatched area represents nonzero polarization, whose value is roughly proportional to the color contrast.

Figure 14

(a) Schematic illustration of the cycloidal structure of the ICM2 phase in zero magnetic field for $\alpha$-${\mathrm{NaFeO}}_2$ and directions of the two electric dipole components in the $ac$ plane, ${\mathit{p}}_1\propto {\mathit{r}}_{ij}\times ({\mathit{S}}_i\times {\mathit{S}}_j)$ and ${\mathit{p}}_2\propto {\mathit{S}}_i\times {\mathit{S}}_j$ allowed by the magnetic symmetry. In a magnetic field, additional (b) cycloid and (c) proper screw modulations along $a$ and $c$ axis are induced, which generate the electric polarizations, ${\mathit{p}}_{3\left(\text{cycloid}\right)}\propto {\mathit{r}}_{ij}\times ({\mathit{S}}_i\times {\mathit{S}}_j)$ and ${\mathit{p}}_{3\left(\text{helix}\right)}\propto [{\mathit{r}}_{ij}·({\mathit{S}}_i\times {\mathit{S}}_j)]\mathit{A}$, respectively.

Figure 15

The spin direction dependence of the magnetic dipole energy. For $-{90}^{\circ }\le \theta \le 0^{\circ }$, $\theta$ corresponds to the angle tilted toward the hexagonal $c$ axis from the $b$ axis, and the angle toward the $a$ axis for $0^{\circ }\le \theta \le {90}^{\circ }$. The calculation was carried out with the system size of 1352 sites.