We have investigated magnetic and dielectric properties of rhombohedral 3R-AgFeO2 and hexagonal 2H-AgFeO2, by using magnetic and dielectric bulk measurements and neutron diffraction experiment with single crystals grown by a hydrothermal synthesis. Magnetic phase transitions occur at T =14.0 K and T =6.0 K in 3R-AgFeO2, and T =17.0 K and T =9.5 K in 2H-AgFeO2 under zero magnetic field. Multistep metamagnetic phase transitions were observed in 3R-AgFeO2 in magnetization measurements up to 60 T, while a single phase transition occurs in 2H-AgFeO2. The ferroelectric polarization parallel and perpendicular to the triangular lattice plane appears below T = 6.0 K in 3R-AgFeO2, which is concomitant with onset of the cycloid magnetic ordering with the propagation vector k = (− 1 2 , q, 1 2 ; q ≃ 0.2) and the magnetic point group polar m1 ′. On the other hand, the ferroelectric polarization is absent even below the lower phase transition temperature in 2H-AgFeO2, which can be explained by the proper screw magnetic structure with k = (0, q, 0; q ≃ 0.4) and the nonpolar 2221 ′ point group. Although the two dimensional triangular lattice layers of 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 cycloid for 3R-AgFeO2 can be explained by the extended inverse Dzyloshinskii-Moriya effect with two orthogonal components, p1 ∝ rij × [Si × Sj] and p2 ∝ Si × Sj. Unlike other delafossite compounds, the p2 component is not allowed in the proper screw phase of 2H-AgFeO2 due to the symmetry restriction of the parent space group.

The magnetic and dielectric properties of the multiferroic triangular lattice magnet compound α-NaFeO 2 were studied by magnetization, specific heat, dielectric permittivity, and pyroelectric current measurements and by neutron diffraction experiments using single crystals grown by a hydrothermal synthesis method. This work produced magnetic field (in the monoclinic ab-plane, B ab , and along the c * -axis, B c ) versus temperature magnetic phase diagrams, including five and six magnetically ordered phases in B ab and along B c , respectively. In zero magnetic field, two spin-density-wave orderings with different k vectors-(0,q, 1 2 ) in phase I and (q a ,q b ,q c ) in phase II-appeared at T = 9.5 and 8.25 K, respectively. Below T = 5 K, a commensurate order with k = (0.5,0,0.5) was stabilized as the ground state in phase III. Both B ab 3 T and B c 5 T were found to induce ferroelectric phases at the lowest temperature (2 K), with an electric polarization that was not confined to any highly symmetric directions in phases IV ab (3.3 B ab 8.5 T), V ab (8.5 B ab 13.6 T), IV c (5.0 B c 8.5 T), and V c (8.5 B c 13.5 T). In phase VI c , within a narrow temperature region in B c , the polarization was confined to the ab plane. For each of the ferroelectric phases, the k vector was (q a ,q b ,q c ), and noncollinear structures were identified, including a general spiral in IV ab an ab cycloid in IV c and V c , and a proper screw in VI c , along with a triclinic 11 ′ magnetic point group allowing polarization in the general direction. Comparing the polarization direction to the magnetic structures in the ferroelectric phases, we conclude that the extended inverse Dzyaloshinskii-Moriya mechanism expressed by the orthogonal components p 1 ∝ r ij × (S i × S j ) and p 2 ∝ S i × S j can explain the polarization directions. Based on calculations incorporating exchange interactions up to fourth-nearest-neighbor (NN) couplings, we infer that competition among antiferromagnetic second NN interactions in the triangular lattice plane, as well as weak interplane antiferromagnetic interactions, are responsible for the rich phase diagrams of α-NaFeO 2 .