Ferroelectricity and lattice distortion associated with spin orderings in a multiferroic delafossite AgFeO2

Spin-lattice coupling and ferroelectric polarization associated with the magnetic ordering in a geometrically frustrated delafossite AgFeO2 have been studied by high resolution neutron powder diffraction and dielectric measurements. The cycloidal magnetic ground state found in this material and the character of the lattice distortions are unprecedented in the family of the delafossite compounds, implying a peculiar mechanism lifting the geometrical frustration.


Introduction
In recent years, physics of geometric frustration in triangular lattice antiferromagnets (TLA) has renewed a great interest due to correlation between frustrated magnetism and structural degrees of freedom as found in a number of multiferroic materials.[1,2] ABO 2 compounds with delafossite or α-NaFeO 2 type of the crystal structure, where A-site is nonmagnetic (Ag, Cu, Pr, Pd, or Na) and B-site is magnetic (Cr, Fe or Ni), [3] have provided a great opportunity to study the frustrated magnetism in TLA.Especially, the cases of CuFeO 2 (pure and doped with Al or Ga) and CuCrO 2 have been intensively studied in last several years due to the spin-driven improper ferroelectricity and spinlattice coupling phenomena.[4][5][6][7][8][9][10] AgFeO 2 belongs to the delafossite ABO 2 family of compounds with rhombohedral R 3m symmetry.Its crystal structure, with the unit cell parameters a = b = 3.0391(1) Å and c = 18.5899(9)Å at room temperature, is shown in Fig. 1.Although the other delafossites, such as CuFeO 2 and CuCrO 2 , have been extensively explored as frustrated and multiferroic magnets,[4-10] AgFeO 2 has not been studied very well, due to the lack of high quality samples.Although one paper has reported the neutron diffraction experiments recently, the magnetic structure was not determined at that time.[11] Very recently, Tsujimoto et al. have succeeded to synthesize AgFeO 2 under high pressure.[12] Using these samples, we performed neutron diffraction and dielectric measurements which revealed two magnetic phase transitions at T N1 = 15 K and T N2 = 9 K. [13] In the temperature range T N2 ≤ T ≤ T N1 (ICM1 phase), the magnetic structure is a spin-density wave with a temperature dependent incommensurate modulation k = ( 1 q 1 2 ).Below T N2 (ICM2 phase), the magnetic structure turns into elliptical cycloid with the incommensurate propagation vector k = ( 1 2 q 1 2 ), giving rise to electric polarization in accordance with the inverse Dzyaloshinskii-Moriya (DM) mechanism.[13] a e-mail: terada.noriki@nims.go.jpIn this paper, we review the magnetic ordering and ferroelectricity as well as we provide the results of high-resolution neutron diffraction experiments revealing the strong magnetoelastic coupling in the frustrated spin system of AgFeO 2 .

Experimental details
The powder sample of AgFeO 2 was prepared under high pressure as described in Ref. [12] High resolution neutron diffraction measurements were carried out at the ISIS facility (UK) using two time-of-flight diffractometers, HRPD [14] to determine the structural parameters, and WISH [15] to analyse the magnetic ordering.The ferroelectric polarization was determined by pyroelectric current measurements with the poling electric field being 800 kV/m.

Magnetic orderings and ferroelectricity
Let us first briefly describe the spin ordering in AgFeO 2 accompanied by structural distortions reducing the rhombohedral symmetry of the paramagnetic phase down to monoclinic.The magnetic reflections indexed by the propagation vector k = ( 1 q 1  2 ) refereeing to the monoclinic cell shown in Fig. 1 (right) have been observed below T N1 = 15 K and successfully modeled in the collinear sinusoidally modulated spin structure.The incommensurate modulation depends on temperature and in the range of stability of the ICM1 phase 9 K ≤ T ≤ 15 K varies as 0.383 ≤ q ≤ 0.389.The deduced centrosymmetric magnetic point group, 2/m1 , is consistent with the observed monoclinic peak splitting (discussed in the next section) and the dielectric measurements testifying the lack of polarization just below the first transition.Both k-vector and magnetic structure are almost the same as in the intermediate temperature phase of CuFeO 2 .[16,17] The spin direction, however, is significantly different in these compounds, suggesting that the A-site cations play important role determining the magnetic anisotropy of the Fe 3+ spins.
Below the second transition at T N2 = 9 K, a cycloidal magnetic structure with propagation vector k = ( 1 2 q 0.205 1  2 ) and elliptical modulation is stabilized.One of the cycloidal axis coincides with the monoclinic b axis and another is almost parallel to the hexagonal c axis.[13] Note that the cycloidal magnetic ground state in AgFeO 2 is completely different from the ground states in other members of the ABO 2 family, in particular, from the collinear ordering in CuFeO 2 and proper screw state in both CuCrO 2 and CuFe 1−x Ga x O 2 .[8,10,18] This implies that the exchange interactions among the spins on the B sites as well as their magnetic anisotropy are significantly affected by the substitution of the nonmagnetic A-site cations.The importance of Cu ions in CuFeO 2 has been pointed out by Malvestuto et al. [19] in the recent X-ray absorption spectroscopy measurements.
intrinsic value of 600 µC/m 2 in a single crystal, which is comparable with the improper polarization in other multiferroics with noncollinear spin ordering.[4] As discussed in our previous paper, [13] the ferroelectric polarization can be understood in terms of the inverse Dzyloshinskii-Moriya mechanism.

Lattice distortion
At high temperature in the paramagnetic phase, the crystal structure of the AgFeO 2 is well described by the rhombohedral R 3m symmetry.The structural parameters refined at T = 20 K are summarized in Table 1 and the quality of the fit is demonstrated in Fig. 4(a).The situation, however, changes below the magnetic phase transition at T N1 = 15 K.The spin ordering results in a clear splitting some of the nuclear peaks as shown in Fig. 3, where the temperature evolution of the (110) reflection is presented.While this reflection is a single peak above T N1 , it separates into two components below this temperature.The splitting progressively increases with decreasing temperature in correlation with the development of the magnetic intensity and the associated order parameter.
In accordance with the 2/m1 symmetry of the spin ordering in the ICM1 phase, which allows the coupling between the magnetic order parameter and the corresponding symmetarized combinations of the macroscopic strain components reducing the crystal structure symmetry down to monoclinic C2/m, the nuclear structure of the low temperature phase has been described using this space group (Table 1).The relation between the hexagonal and monoclinic cells are shown in Fig. 1 (right).It should be pointed out that the incommensurate nature of the magnetic ordering necessarily implies the existence of structural modulations with the propagation vectors 2nk magnetic (n is integer).However, the corresponding satellites were not observed in the present powder diffraction experiments and therefore only coupling with macroscopic quantities was taken into account. 15008-p.3

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In the ICM2 phase, in addition to the monoclinic strains, a macroscopic polarization is coupled to the lattice distortions due to the polar character of the cycloidal magnetic ordering.Attempts to refine the crystal structure using the polar Cm space group did not result in any improvement of the fitting quality and therefore the centrosymmetric C2/m space group was used over all temperatures in both ICM1 and ICM2 phases (Fig. 4(b),(c)).
In spite of the common low-temperature monoclinic symmetry in both AgFeO 2 and CuFeO 2 delafossites, the quantitative character of their lattice distortions is essentially different.In particularly, the b axis in the monoclinic setting contracts below the magnetic phase transition in AgFeO 2 and elongates in CuFeO 2 , implying different mechanisms of releasing the spin frustration in these compounds.

Summary
We have studied the magnetic ordering, ferroelectricity, and crystal lattice distortions in the multiferroic delafossite AgFeO 2 .Two magnetic phase transitions at T N1 = 15 K and T N2 = 9 K have been revealed and the corresponding magnetic and crystal structures were successfully refined.In the temperature range 9 K ≤ T ≤ 15 K, the magnetic structure is a non-polar (2/m1 ) spin density wave with the incommensurate propagation vector k = ( 1 q 1  2 ) being dependent on temperature.Below 9 K, the magnetic structure turns into the polar (m1 ) elliptical cycloid with k = ( 1 2 q 1 2 ), giving rise to the macroscopic polarization.Analysis of the high resolution neutron diffraction data revealed a clear lattice distortions associated with the onset of the magnetic ordering.The deduced monoclinic symmetry is consistent with the symmetry of the magnetic order parameter and indicates a strong coupling of the latter with the macroscopic strains.The quantitative analysis of the lattice distortions points to a different mechanism lifting the geometrical frustration in AgFeO 2 than in the case of the well studied CuFeO 2 delafossite.

Fig. 2 .
Fig. 2. Temperature dependence of (a) magnetic neutron intensity measured on WISH diffractometer and (b) ferroelectric polarization in AgFeO 2 .Closed and open symbols denote data on heating and cooling, respectively.Dotted lines show the magnetic phase transition temperatures.

Fig. 3 .
Fig. 3. Temperature dependence of the hexagonal 110 nuclear diffraction profile of AgFeO 2 .The data were measured on HRPD at increasing temperature.Below T N1 = 15 K, the peak is split into two components indexed as 020 and 111 in the monoclinic cell.

Fig. 4 .
Fig. 4. Typical results of the Rietveld refinement of the experimental data collected on HRPD at (a) 20 K, (b) 10 K and (c) 4 K.The refinements were performed with FullProf program.[20]The vertical bars indicate nuclear (upper row) magnetic (bottom row) Bragg peaks.The refined parameters and reliability factors are listed in Table1.