Crystal structure of solid solutions REFe_1−x(Al or Ga)_xO₃ (RE=Tb, Er, Tm) and the correlation between superexchange interaction Fe⁺³–O⁻²–Fe⁺³ linkage angles and Néel temperature

Abstract

The T_N temperatures of rare earth orthoferrites depend on magnetic interactions as well as on the details of the REFeO₃ structure. The magnetic interactions in this structure can be easily modified by substitution of Fe⁺³ ions by non-magnetic ones. The paper shows that in solid solutions of rare earth orthoferrites with correspondent orthoaluminates and orthogalates (REFe_1−xAl_xO₃, REFe_1−xGa_xO₃), similar as in orthoferrites, a close dependence of the T_N temperature on the geometry of the system can be observed. On the basis of this dependence, the Néel temperature (767.1±3.7 K) of a hypothetical orthoferrite with 180° angle of Fe⁺³–O⁻²–Fe⁺³ linkage has been determined.

Introduction

Rare earth orthoferrites REFeO₃ (RE=La…Lu and Y) create a family of compounds, which crystallise in an orthorhombically distorted perovskite structure belonging to the Pbnm (D_2h¹⁶) space group. The elementary unit cell consists of four molecules. The RE⁺³ cations occupy nearly central positions (Wyckoff position-4(c)) in a parallelepiped formed by Fe⁺³ (4(b)) cations. The iron ions are nearly octahedrally coordinated with six O⁻². Two of the O⁻² occupies positions 4(c), and the other four occupy positions 8(d) (Table 1 and Fig. 1). The O⁻²(4c)–O⁻²(4c) axes of distorted octahedra are tilted alternately, forming a zigzag along the [0 0 1] direction with the tilt angle γ

$$\text{γ=}\text{cos}{}^{\mathrm{-1}}\text{c}\text{4p}{}_0\text{,p}{}_0\text{=}\text{x}{}_{\mathrm{<mtext>O</mtext><mtext>(4</mtext><mtext>c</mtext><mtext>)</mtext>}}{}^2\text{a}{}^2\text{+}\text{1}\text{2}\text{−y}{}_{\mathrm{<mtext>O</mtext><mtext>(4</mtext><mtext>c</mtext><mtext>)</mtext>}}{}^2\text{b}{}^2\text{+}\text{1}\text{4}{}^2\text{c}{}^2\text{.}$$

The dihedral angle ($\text{α=2}\text{tan}{}^{\mathrm{-1}}\text{a/b}$) in the Fe⁺³ parallelepiped can be treated as a measure of the orthorhombic distortion of the perovskite structure in the ab plane. As the difference of lattice constants a and b approaches zero, the α angle approaches 90°, which means a decrease in the degree of distortion.

As the radius of the RE⁺³ cations located in deformed dodecahedral interstices between the oxygen octahedra increases (Lu⁺³(0.97 Å)→La⁺³(1.18 Å) [1]), all the ions apparently shift in such a way that lattice size increases [2], [3] (Fig. 2) and the distortion degree as well the tilt angle decrease (Fig. 3).

Among all the possible magnetic interactions in the REFeO₃ orthoferrites, the superexchange coupling Fe⁺³–O⁻²–Fe⁺³ of iron magnetic moments is dominant and leads to a non-colinear antiferromagnetic (weak ferromagnetic) Fe⁺³ sublattice ordering. The direct Fe⁺³–Fe⁺³ coupling is much weaker, and magnetic RE⁺³–Fe⁺³ and RE⁺³–RE⁺³ couplings are of the 10⁻²–10⁻³ orders in relation to the Fe⁺³–O⁻²–Fe⁺³ ones [4], [5].

Phenomenologically, the Néel temperature T_N of the antiferromagnetically ordered iron sublattice is proportional to the average number (Z=6) of linkages per Fe⁺³ ion, to the exchange constant ($\text{I}\text{<0}$) of Fe⁺³ ion pairs and to the average cosine of the Θ angle of the Fe⁺³–O⁻²–Fe⁺³ linkage [4], [6]

$$\text{T}{}_{\mathrm{<mtext>N</mtext>}}\text{∼}\text{I}\text{ZS(S+1)}\text{cos}\text{Θ,}$$

where $\text{S=}\text{5}\text{2}$ for Fe⁺³, and $\text{I}$ is defined by the Hamiltonian $\text{H}\text{=−}\text{I}\text{∑}\text{〈i,j〉}\text{S}{}_{\mathrm{i}}\text{·}\text{S}{}_{\mathrm{j}}$ [4].

The above formula (2) can be written [4], [5] as

$$\text{T}{}_{\mathrm{<mtext>N</mtext>}}\text{=−T}{}_{\mathrm{<mtext>N</mtext>}}\text{(Θ=180°)}\text{cos}\text{Θ,}$$

where T_N(Θ=180°) ($\text{∼−}\text{I}\text{ZS(S+1)}$) coefficient is the Néel temperature of a hypothetical orthoferrite with the Fe⁺³–O⁻²–Fe⁺³ linkage angle of 180°.

An interesting question concerns the validity limits of that simple relation (3). A decrease in the exchange interaction can easily be achieved when the iron sublattice is diluted with non-magnetic atoms, as in the case of solid solutions of rare earth orthoferrites with correspondent orthoaluminates or orthogalates (REF_1−xAl_xO₃, REFe_1−xGa_xO₃). These compounds are, therefore, a suitable material for verifying the above mentioned dependence.