Magnetoelectric properties of A₂[FeCl₅(H₂O)] with A = K, Rb, Cs

The magnetic-susceptibility measurements of A₂[FeCl₅(H₂O)] with A = K, Rb, Cs are summarized in figure 2. The magnetic properties of all three compounds are rather similar. The low-field curves χ(T) show well-defined kinks at $T_{\rm N}^{\rm{K}}\simeq 14.3\,{\rm K}$, $T_{\rm N}^{\rm{Rb}}\simeq 10.2\,{\rm K}$ and $T_{\rm N}^{\rm{Cs}}\simeq 6.8\,{\rm K}$ for A = K, Rb and Cs, respectively, in agreement with previous results [6, 10, 12, 17–19]. In all three cases, χ_a decreases below T_N with decreasing temperature and approaches zero for T → 0 K, while χ_b and χ_c hardly change, indicating that the spins are oriented parallel to the a axis, the magnetic easy axis. Note, however, that with respect to the zigzag chains of [FeCl₅(H₂O)]²⁻ octahedra the a axis is oriented differently in the two structure types, see figure 1. Therefore, the magnetic easy axis in the Cs-based compound is perpendicular to that in the K-based and Rb-based compound. In the K-based and Rb-based compound $T_{\rm N}^{\rm{K}}$ and $T_{\rm N}^{\rm{Rb}}$ decrease slightly with increasing magnetic field independent of the field direction. In contrast, in the Cs-based compound $T_{\rm N}^{\rm{Cs}}$ strongly decreases for all three field directions with increasing magnetic fields, which is in agreement with the results of a Mössbauer study [20].

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For larger magnetic fields parallel to a, the decrease of χ_a(T) below T_N systematically vanishes for all compounds. Above the critical fields $B_{\rm{SF}}^{\rm{K}}\simeq 3.5\,{\rm T}$, $B_{\rm{SF}}^{\rm{Rb}}\simeq 1.5\,{\rm T}$ and $B_{\rm{SF}}^{\rm{Cs}}\simeq 1.2\,{\rm T}$ the characteristics of χ_i(T) become nearly identical in all the compounds for all three magnetic-field directions a, b, c, see figures 2(a), (d) and (i). Therefore, spin-flop transitions occur at $B_{\rm{SF}}^{A}$, with a rotation of the spins from being oriented along the a axis to lying within the plane that is perpendicular to a. These spin-flop transitions can also be seen in the low-temperature measurements of the magnetization as a function of the magnetic field, see the insets in figures 2(a), (d) and (i). Note that the field-dependent magnetization of the K-based compound was obtained by resorting the temperature-dependent magnetization data.

The inverse susceptibilities follow a Curie–Weiss behaviour from about 50 K up to room temperature for all three compounds, see the insets in figures 2(b), (e) and (g). Here, only the data of 1/χ_b are displayed, because the data of 1/χ_a and 1/χ_c for all the compounds are almost identical in the high-temperature regime. Linear fits to the data of $1/\chi_i^{A}$ for T > 100 K yield negative Weiss temperatures θ^A ≃ −39 K, −31 K, −12 K for A = K, Rb and Cs, respectively, and an effective magnetic moment μ_eff ≃ 6 μ_B. The negative Weiss temperatures signal net antiferromagnetic exchange interactions, whose magnitude roughly scales with T_N. With increasing ionic radius of the alkali ions, the sum of the exchange interactions diminishes, most probably due to the increasing distance of the magnetic ions. The value of the effective magnetic moment lies close to $\mu_{\rm{eff}}=g\mu_{\rm B}\sqrt{S(S+1)}=5.92$ μ_B, as expected for a 3d⁵ high-spin configuration of the Fe³⁺ ions and g = 2.

Figure 3 displays the temperature-dependent measurements of the electric polarization P = (P_a, P_b, P_c) of A₂[FeCl₅(H₂O)] with A = K, Rb and Cs in magnetic fields applied parallel to a, b and c. An electric polarization does not arise in any of the three compounds below the corresponding Néel temperature as long as there is no magnetic field applied. Applied magnetic fields, however, induce an electric polarization in all cases. Therefore, the three compounds are not ferroelectric (multiferroic) but show a magnetoelectric effect. The magnetic-field configurations, where no electric polarization is induced, are not presented here. The induced components of the electric polarization of K₂[FeCl₅(H₂O)] and Rb₂[FeCl₅(H₂O)] behave very similarly with respect to their temperature and magnetic-field dependences. The induced electric polarization of Cs₂[FeCl₅(H₂O)] has slightly different characteristics.

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As can be seen in figures 3(a) and (d), the polarization components $P_a^{\rm{K}}$ and $P_a^{\rm{Rb}}$ in the K-based and Rb-based compound reach broad maxima around 12 K and 9 K, respectively, for B||a and then approach zero for T → 0 K. With increasing field strength $P_a^{\rm{K}}$ and $P_a^{\rm{Rb}}$ grow linearly in the low-field range until they are suppressed completely in the vicinity of the spin-flop transitions at $B_{\rm{SF}}^{\rm{K}}=3.5\,{\rm T}$ and $B_{\rm{SF}}^{\rm{Rb}}=1.5\,{\rm T}$, respectively. At higher field strengths, electric polarizations P_c parallel to c occur, which also grow linearly with further increasing magnetic fields, see figures 4(a) and (b). For both compounds, the temperature dependences of P_b and P_c for B||b and B||c, respectively, are very similar, but different from P_a for B||a, see figures 3(b), (c), (e) and (f). In both cases the induced polarizations again depend linearly on the field strength and P_c is about one order of magnitude larger than P_b.

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In Cs₂[FeCl₅(H₂O)], a magnetic field parallel to a induces an electric polarization along b, while a magnetic field parallel to b causes an electric polarization along a, see figures 3(g) and (h). The temperature dependence of $P_b^{\rm{Cs}}$ for B||a is very similar to that of $P_a^{\rm{K}}$ and $P_a^{\rm{Rb}}$ for B||a. It reaches a broad maximum around 5 K and then approaches zero for T → 0 K. In the vicinity of the spin-flop transition at $B_{\rm{SF}}^{\rm{Cs}}=1.2\,{\rm T}$, $P_b^{\rm{Cs}}$ is suppressed completely and instead an electric polarization with the components $P_a^{\rm{Cs}}$ and $P_c^{\rm{Cs}}$ arises, see figures 4(c) and (d). For B||b an electric polarization along a is created, which saturates quickly with decreasing temperature, and for B||c no electric polarization emerges at all.

The occurrence of the magnetic-field induced electric polarizations below the Néel temperature T_N also causes anomalies in the temperature and magnetic-field dependences of the corresponding longitudinal components $\epsilon_i^r$ of the dielectric tensor. To illustrate this, representative measurements of the temperature-dependent $\epsilon_i^r$ (i = a, b, c) of K₂[FeCl₅(H₂O)] for B parallel to a, b and c (at a frequency of 1 kHz) are displayed in figure 5. The zero-field curves show only faint kinks at T_N, but in finite magnetic fields, spiky anomalies occur at the transition temperatures, which grow in intensity with increasing magnetic field in all cases. For B||a, above the magnetic spin-flop field, the spiky anomaly disappears again, while for the other field directions the anomalies stay present up to the maximum field strength of 15 T. In the vicinity of the spin-flop field, a second anomaly occurs, which coincides with the suppression of the electric polarization component P_a.

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For magnetic fields well below or above the spin-flop transitions, the magnetic-field induced electric polarization in A₂[FeCl₅(H₂O)] with A = K, Rb, Cs in all cases depends linearly on the magnetic field strength. This indicates that all three compounds are linear magnetoelectrics. The linear field dependence of the electric polarization of all compounds is illustrated in figures 6 and 7, by plotting the components $P_i^{\rm{K}}$, $P_i^{\rm{Rb}}$ and $P_i^{\rm{Cs}}$ (i = a, b, c) as functions of the magnetic field at representative temperatures. Linear fits to the magnetic-field dependent electric polarization data yield the linear magnetoelectric tensor components α_ij(T) = ∂μ₀P_i(T)/∂B_j in SI units.

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The linear magnetoelectric tensors of K₂[FeCl₅(H₂O)] and Rb₂[FeCl₅(H₂O)] have a diagonal form with the non-zero components α₁₁, α₂₂ and α₃₃ (related to unit vectors e₁, e₂, e₃ parallel to the crystallographic axes a, b, c, respectively). This result is compatible with the magnetic point-group symmetry m'm'm', determined by neutron-diffraction measurements [12, 13]. The temperature dependences and absolute values of the corresponding tensor components α_ii(T) of both compounds are very similar, as can be seen in figures 8(a) and (d). Below the Néel temperatures, α₂₂(T) and α₃₃(T) grow strongly and then saturate with decreasing temperature, while α₁₁(T) first reaches a broad maximum and finally approaches zero for T → 0. In contrast to the K-based and Rb-based compounds, Cs₂[FeCl₅(H₂O)] has only off-diagonal tensor components α₂₁ and α₁₂ for B < B_SF. The component α₂₁(T) has an analogous temperature dependence as $\alpha_{11}^{\rm K,Rb}(T)$ and the component α₁₂(T) behave similar to $\alpha_{22}^{\rm K,Rb}(T)$ and $\alpha_{33}^{\rm K,Rb}(T)$, see figure 8(g). The linear magnetoelectric tensors of A₂[FeCl₅(H₂O)] with A = K, Rb, Cs calculated at representative temperatures are displayed in table 3.

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Table 3. Linear magnetoelectric tensors of A₂[FeCl₅(H₂O)] with A = K, Rb, Cs at representative temperatures. The tensors in the right column refer to the spin-flop phases for B > B_SF along a, where only the components α_i1 could be determined.

Compound	T (K)	[αij]AF (ps/m)	[αij]SF (ps/m)
K2[FeCl5(H2O)]	11 K	$\left[\begin{array}{ccc} 0.71 & \sim 0 & \sim 0\\ \sim 0 & 0.08 & \sim 0\\ \sim 0 & \sim 0 & 1.17 \end{array}\right]$	$\left[\begin{array}{ccc} \sim 0 & \,\quad\mbox{---}\quad & \mbox{---}\\ \sim 0 & \mbox{---} & \mbox{---}\\ 0.71 & \mbox{---} & \mbox{---}\end{array}\right]$
Rb2[FeCl5(H2O)]	8 K	$\left[\begin{array}{ccc} 0.63 & \sim 0 & \sim 0\\ \sim 0 & 0.02 & \sim 0\\ \sim 0 & \sim 0 & 0.98 \end{array}\right]$	$\left[\begin{array}{ccc} \sim 0 & \quad\mbox{---}\quad & \mbox{---}\\ \sim 0 & \mbox{---} & \mbox{---}\\ 1.0 & \mbox{---} & \mbox{---}\end{array}\right]$
Cs2[FeCl5(H2O)]	5 K	$\left[\begin{array}{ccc} \sim 0 & 0.45 & \sim 0\\ 3.35 & \sim 0 & \sim 0\\ \sim 0 & \sim 0 & \sim 0\end{array}\right]$	$\left[\begin{array}{ccc} 0.03 & \quad\mbox{---}\quad &\mbox{---}\\ \sim 0 & \mbox{---}& \mbox{---}\\ 0.02 & \mbox{---} &\mbox{---}\end{array}\right]$

The spin-flop transitions for B||a lead to a modification of the magnetoelectric responses in all A₂[FeCl₅(H₂O)] compounds. For A = K or Rb, the component α₁₁ is suppressed and α₃₁ occurs. For A = Cs, the component α₂₁ is suppressed and α₁₁ as well as α₃₁ occur. Their temperature dependences are very similar to each other, see figures 8(c), (f) and (i). The behaviour of the other tensor components in the spin-flop phases could not be determined, because in the experimental setup used, it was not possible to apply a constant magnetic field B > B_SF parallel to a, while applying a variable magnetic field parallel to b or c.

For collinear antiferromagnetic magnetoelectrics, the temperature dependences of the linear magnetoelectric tensor components for B parallel (α_||) and B perpendicular (α_⊥) to the magnetic easy axis are characterized by the product of the corresponding magnetic susceptibilities χ_||(T) and χ_⊥(T) with the sublattice magnetization $\bar{S}(T)$ [14, 15] via

$$\begin{eqnarray} &&\fl \alpha_{||}(T) \propto \chi_{||}(T)\cdot\bar{S}(T), \quad \alpha_{\perp}(T) \propto \chi_{\perp}(T)\cdot\bar{S}(T). \end{eqnarray} \tag{ 1 }$$

As is shown in figure 8 the determined temperature-dependent α_ij(T) agrees well with the expected behaviour according to equation (1). For magnetic fields perpendicular to the magnetic easy axis (B||b or B||c), the temperature dependences of the magnetoelectric responses are dominated by that of the sublattice magnetization $\bar{S}$, because the corresponding magnetic susceptibilities stay nearly constant below T_N, see figure 2. Therefore, α₂₂(T), α₃₃(T) for the K-based and Rb-based compound and α₁₂(T) for Cs₂[FeCl₅(H₂O)] have the form of an order parameter. For magnetic fields parallel to the magnetic easy axis (B||a), the temperature dependences of the magnetoelectric responses are dominated by the corresponding magnetic susceptibilities, which approach zero for T → 0. Therefore, α₁₁(T) for the K-based and Rb-based compound and α₂₁(T) for Cs₂[FeCl₅(H₂O)] approach zero as well for T → 0. The different temperature characteristics of the magnetoelectric responses are further illustrated in figures 8(b), (e) and (h), where the ratios α_||/χ_|| and α_⊥/χ_⊥ are displayed. For the calculation of the ratios α/χ, the magnetic volume susceptibilities in SI units χ^SI = 4π χ^CGS,mol/V_mol are used. In most of the cases the expected temperature dependence of an order parameter is nicely reproduced. Partially, slight deviations occur in the low-temperature limit. These can be explained by problems in determining the accurate absolute values of the electric polarization from the results of pyroelectric-current measurements, when the respective pyroelectric currents hardly exceed the background currents.

As mentioned in the introduction, the magnetic structure of Cs₂[FeCl₅(H₂O)] is still unknown. With the knowledge of the anisotropy of the linear magnetoelectric effect of Cs₂[FeCl₅(H₂O)] it is, however, possible to deduce its magnetic space group by the following symmetry analysis.

Starting with the point group symmetry $\frac{2}{m}\frac{2}{m}\frac{2}{m}$ for the prototypic, paramagnetic phase of Cs₂[FeCl₅(H₂O)], the only possible magnetic point groups, compatible with the form of the linear magnetoelectric tensor, are mm2, 2'mm', 2'2'2 and mmm'. The magnetic susceptibility measurements reveal that the spins are ordered antiferromagnetically below the Néel temperature with a as the magnetic easy axis. In addition, the results of the dielectric investigations show that in the antiferromagnetically ordered phase, no electric polarization occurs in zero magnetic field. Therefore, the two polar groups mm2 and 2'mm' can be (almost certainly) excluded, as well as the ferromagnetic group 2'2'2. Hence, the only remaining possibility is the magnetic point group $mmm^{\prime}\mathop{\hat=}\frac{2^{\prime}}{m}\frac{2^{\prime}}{m}\frac{2}{m^{\prime}}$. Including the information about the space group $Cmcm\mathop{\hat=}C\frac{2}{m}\frac{2}{c}\frac{2_1}{m}$ of the room-temperature crystal structure of Cs₂[FeCl₅(H₂O)], the (only possible) magnetic space group for its linear magnetoelectric phase is $C\frac{2^{\prime}}{m}\frac{2^{\prime}}{c}\frac{2_1}{m^{\prime}}$, which corresponds to the magnetic space group number 63.5.515 according to [16]. With this knowledge, a model for the magnetic structure of Cs₂[FeCl₅(H₂O)] can be derived. Starting with a spin at an arbitrary Fe site oriented along a, the orientations of all the other spins within the structure follow by applying all the symmetry elements of the magnetic space group $C\frac{2^{\prime}}{m}\frac{2^{\prime}}{c}\frac{2_1}{m^{\prime}}$. In figure 9 the resulting magnetic structure of Cs₂[FeCl₅(H₂O)] below T_N = 6.8 K is displayed for one unit cell. The spin direction alternates between ±a along the zigzag chains of the [FeCl₅(H₂O)]²⁻ octahedra running along c and the spins of neighbouring chains are in phase. Therefore, the magnetic structure of the Cs-based compound is the same type as that of the K-based and Rb-based compounds. The direction of the AFM zigzag chains is equivalent in the Pnma-type and Cmcm-type structure, but the orientations of the magnetic easy axes differ by 90°.

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