Successive Short- and Long-Range Magnetic Ordering in Ba₂Mn₃(SeO₃)₆ with Honeycomb Layers of Mn³⁺ Ions Alternating with Triangular Layers of Mn²⁺ Ions

Abstract

Mixed-valent Ba₂Mn²⁺Mn₂³⁺(SeO₃)₆ crystallizes in a monoclinic P2₁/c structure and has honeycomb layers of Mn³⁺ ions alternating with triangular layers of Mn²⁺ ions. We established the key parameters governing its magnetic structure by magnetization M and specific heat C_p measurements. The title compound exhibits a close succession of a short-range correlation order at T_corr = 10.1 ± 0.1 K and a long-range Néel order at T_N = 5.7 ± 0.1 K, and exhibits a metamagnetic phase transition at T < T_N with hysteresis most pronounced at low temperatures. The causes for these observations were found using the spin exchange parameters evaluated by density functional theory calculations. The title compound represents a unique case in which uniform chains of integer spin Mn³⁺ (S = 2) ions interact with those of half-integer spin Mn²⁺ (S = 5/2) ions.

1. Introduction

Compounds of transition metal magnetic ions exhibit a wide range of phenomena, which are commonly grouped in terms of their spin values: quantum magnetism for systems of small spins and classical magnetism for systems of large spins. In both groups, the ground state that a magnetic compound reaches at low temperatures is governed largely by the dimensionality of its spin exchange interactions and also by the degree of geometrical spin frustration (hereafter, spin frustration) [1]. It quickly becomes complicated to analyze the properties of a magnetic system with magnetic ions of two (or more) different oxidation states and two (or more) different values of spins in terms of these two factors unless the relative values of its various spin exchanges are known. These days, it has become almost routine to determine the values of spin exchanges for any complex magnetic system by performing energy mapping analysis [2,3,4]. This method employs a number of broken-symmetry spin states of a given magnetic system, then evaluates their energies using a spin Hamiltonian made up of the spin exchanges, in addition using density functional theory (DFT) calculations and finally maps the relative energies of the broken-symmetry states from the spin Hamiltonian to those of the DFT calculations. In other words, this method relates the energy spectrum of a model Hamiltonian to that of an electronic Hamiltonian using a set of broken-symmetry states. Over the years, the energy mapping analysis has led to correct spin lattice models with which to understand the properties of numerous magnetic materials.

Among low-dimensional mixed-valent compounds of small spin, LiCu₂O₂ attracts attention as a type-II multiferroic since the formation of the long-range spiral magnetic order is accompanied by the emergence of ferroelectricity [5,6]. Another low-spin mixed-valent compound is NaV₂O₅ which undergoes a charge ordering transition, which gives rise to a spin gap in the magnetic excitation spectrum [7]. Among low-dimensional mixed-valent compounds of large spin, manganese-based systems are most extensively studied. Two magnetic orderings and a spin–flop transition were observed in the mixed-valent compound Mn₃O(SeO₃)₃, the magnetic ion arrangement of which shows the intersection of octa-kagomé spin sublattices and staircase-kagomé spin sublattices [8]. The hollandite-type compound, Ba_1.2Mn₈O₁₆, undergoes a magnetic transition at 40 K, which is significantly lower than the Weiss temperature of −385 K, a characteristic feature of high spin frustration. Strong spin frustration usually results from triangular arrangements of magnetic ions with antiferromagnetic spin exchange for each edge and/or competing ferromagnetic and antiferromagnetic interactions [9].

In the present work, we examine how to understand the magnetic properties of the mixed-valent manganese compound Ba₂Mn²⁺Mn³⁺₂(SeO₃)₆ [10]. In general, spin exchanges between transition-metal magnetic ions M forming ML_n polyhedra with surrounding main-group ligands L are classified into M-L-M and M-L…L-M types [2,3,4], and the latter are further differentiated depending on whether or not the L…L bridge is coordinated by a metal cation A^m+ to form L… A^m+…L. In cases when a metal cation is present, the M-L…A^m+…L-M exchanges are further differentiated by whether or not the A^m+ cation is a d⁰ transition-metal cation or a main-group cation. The spin exchanges involving a main-group cation are almost impossible to predict using simple qualitative arguments, especially when this cation makes strong covalent bonds with the ligand to form a molecular anion, e.g., a P₂S₆⁴⁻ anion in MPS₃ (M = Mn, Fe, Co, Ni) [11]. Such is the case for the SeO₃²⁻ anion of Ba₂Mn₃(SeO₃)₆. As will be described below, the spin exchanges of Ba₂Mn₃(SeO₃)₆ are all of the Mn-O…Se⁴⁺…O-Mn type; that is, none involves the Mn-O-Mn type so that the Mn³⁺ and Mn²⁺ ions of Ba₂Mn₃(SeO₃)₆ do not generate double exchanges [12]. The Mn²⁺ ions of Ba₂Mn₃(SeO₃)₆ form chains, with the Mn³⁺ ions in the same direction. Between these chains, triangular arrangements of the magnetic ions such as (Mn²⁺, Mn²⁺, Mn³⁺) and (Mn²⁺, Mn³⁺, Mn³⁺) occur in all three directions, so one might consider the presence of strong spin frustration and exhibit magnetic properties expected for a low-dimensional antiferromagnetic material. To interpret these seemingly puzzling aspects of Ba₂Mn₃(SeO₃)₆, we determine its spin exchanges using the energy-mapping analysis to find the cause for its low-dimension antiferromagnetic behavior.

2. Materials and Methods

Mixed-valent Ba₂Mn²⁺Mn₂³⁺(SeO₃)₆ was synthesized by a hydrothermal reaction of BaCO₃ (2 mmol), MnCl₂∙4H₂O (1 mmol) and H₂SeO₃ (3 mmol) as precursors with 1.5 mL of 65% HNO₃ and 3 mL of water added. The mixture was placed into a Teflon chamber of a steel autoclave (10 mL) after the degassing was finished. The autoclave then was placed into the furnace, where the temperature was raised to 200 °C for 1 week. After this, the brown powder of Ba₂Mn₃(SeO₃)₆ was rinsed with water to wash out the contaminants. The obtained powder sample was found to crystallize in the monoclinic P2₁/c space group with a = 5.4717(3) Å, b = 9.0636(4) Å, c = 17.6586(9) Å, β = 94.519(1), V = 873.03(8) ų, Z = 2 in agreement with the original solution [10] and its powder XRD pattern (BRUKER D8 Advance diffractometer Cu Kα, λ = 1.54056, 1.54439 Å, LYNXEYE detector) is shown in Figure 1.

As depicted in Figure 2a, the structure is composed of MnO₆ octahedra interlinked with SeO₃ pyramids. The Mn1O₆ octahedra of Mn²⁺ (S = 5/2) ions form trigonal layers, and Mn2O₆ octahedra of Mn³⁺ (S = 2) ions form honeycomb layers, as shown in Figure 2b,c.

Physical properties of Ba₂Mn₃(SeO₃)₆ were characterized by measuring the magnetization M and the specific heat C_p on ceramic samples (well-pressed pellets of 3 mm in diameter and 1 mm in thickness) using various options of “Quantum Design” Physical Properties Measurements System PPMS—9 T taken in the range 2–300 K under magnetic field µ₀H up to 9 T.

3. Results

3.1. Magnetic Susceptibility

The magnetic susceptibility χ = M/H of Ba₂Mn₃(SeO₃)₆ taken at µ₀H = 0.1 T in the field-cooled regime is shown in Figure 3. In the high-temperature region, it follows the Curie–Weiss law:

$$\chi =\frac{C}{T-\theta }+{\chi }_0$$

(1)

with the temperature-independent term ${\chi }_0$ = −7.6 × 10⁻⁴ emu/mol, the Curie constant C = 10.98 emu K/mol and the Weiss temperature Θ = −27.8 K. The value of ${\chi }_0$ exceeds the sum of the Pascal constants of ions and groups constituting the title compound ${\chi }_{0,calc}$ = −3.5 × 10⁻⁴ emu/mol [13]. This should be attributed to the effect of sample holder. The value of C somewhat exceeds the value C_calc = 10.375 emu K/mol expected under the assumption of g-factor, g = 2, for both the Mn²⁺ and Mn³⁺ ions. Use of g = 2 is reasonable for Mn²⁺ (S = 5/2) ions with no orbital-moment contribution, but it underestimates the g-factor for Mn³⁺ ions (S = 2). The negative value of the Weiss temperature Θ points to the predominance of antiferromagnetic exchange interactions at elevated temperatures. Its absolute value can be influenced by the competition of ferromagnetic and antiferromagnetic exchange interactions.

On lowering the temperature, the χ(T) curve passes through a broad maximum at T_corr = 10.2 K and shows a kink at T_N = 5.6 K, which is more pronounced in the Fisher’s specific heat d(χT)/dT (not shown). This broad maximum is typically found for a quasi-one-dimensional (1D) antiferromagnetic chain system; hence, suggesting that Ba₂Mn₃(SeO₃)₆ has a 1D-like antiferromagnetic subsystem. The kink at a lower temperature shows that Ba₂Mn₃(SeO₃)₆ undergoes a long-range antiferromagnetic order. The drop of magnetic susceptibility χ below its largest value at T_corr is less than one-third of that expected for a three-dimensional easy-axis antiferromagnet [14]. The absence of a so-called Curie tail at lowest temperatures signals the high chemical purity of the sample.

3.2. Field Dependence of Magnetization

The field dependencies of the magnetization M taken at selected temperatures in the T < T_N and T_N < T < T_corr regions are shown in Figure 4. At the highest temperature of our measurement, the M(H) curve is linear indicating that the system is in the paramagnetic state, but starts to deviate from linearity as the temperature is lowered toward T_N. Below T_N, the M(H) curves exhibit hysteresis, which becomes most pronounced at the lowest temperature of our measurement. In general, the Heisenberg magnets of quasi-isotropic magnetic moment experience a spin–flop transition prior to the full saturation at the spin–flip transition. This is not the case for Ba₂Mn₃(SeO₃)₆, although it has isotropic Mn²⁺ ions. Instead, it exhibits a metamagnetic transition inherent to the Ising magnets. Such behavior should be associated with the presence of highly anisotropic Mn³⁺ ions in the system. The well-pronounced hysteresis underlines the first-order nature of the metamagnetic transition [15].

3.3. Heat Capacity

The magnetization data are fully consistent with the specific heat data, shown in Figure 5. In a wide temperature range, the C_p(T) curve can be described by the sum of the Debye function with Θ_D = 223 K and two Einstein functions with Θ_E1 = 556 K and Θ_E2 = 1449 K. The first of the Einstein functions can be ascribed to the oscillation mode of the MnO₆ octahedra and the second one to that of the SeO₃ pyramids. These parameters were obtained by fitting the data in the 70–290 K region with the fixed sum of the Debye and Einstein functions. The remaining data were considered as a purely magnetic contribution. Indeed, the magnetic entropy is nearly equal to the theoretical value of R(2ln(5) + ln(6)) = 41.6 J/mol K, confirming the accuracy of the fit. Nevertheless, a nonmagnetic analogue is still needed to obtain more accurate values of Debye and Einstein temperatures.

On lowering the temperature, the specific heat C_p passes through a broad maximum at T_corr = 10 K and shows a peak at T_N = 5.8 K. Under external magnetic field, the broad maximum retains its position, but the sharp anomaly shifts to lower temperatures. Such behavior is typical of low-dimensional antiferromagnets experiencing successive short-range and long-range orders.

3.4. Spin Exchanges and Interpretation

The two important issues concerning the observed magnetic properties of Ba₂Mn₃(SeO₃)₆ are the cause for the broad maximum of the magnetic susceptibility at T_corr = 10.1 ± 0.1 K, suggesting a short-range correlation as found for a 1D antiferromagnetic chain and a sharp kink at 5.7 ± 0.1 K, suggesting a long-range antiferromagnetic ordering. With Θ = −27.8 K and T_N = 5.7 K (the index of spin frustration f = 5.0), the spin frustration in Ba₂Mn₃(SeO₃)₆ is not strong enough to prevent it from adopting a long-range antiferromagnetic ordering. This is somewhat surprising because one might expect a strong spin frustration in Ba₂Mn₃(SeO₃)_6. Figure 2c shows that the interaction between a chain of Mn²⁺ ions with the Mn³⁺ ions in the surrounding hexagonal prism generates numerous spin exchange triangles, which is a common arrangement leading to spin-frustration. Furthermore, these interactions must give rise to a 1D antiferromagnetic chain behavior to explain the 1D-like short range correlation at 10.1 K. To explore these issues, we first evaluate the spin exchanges of various exchange paths J in Ba₂Mn₃(SeO₃)₆.

All adjacent magnetic ions of Ba₂Mn₃(SeO₃)₆ are interconnected by the SeO₃ pyramids except for the Mn³⁺ ions encircled by dashed ellipses in Figure 2a. The O…O contact distances (3.988 Å) of their Mn-O…O-Mn exchange paths are well beyond the van der Waals distance of ~3.30 Å so these spin exchanges can be neglected. There are still numerous spin exchanges between the Mn²⁺ and Mn³⁺ ions as depicted in Figure 6a. The spin exchange paths J₂ (J₁) form chains of Mn³⁺ (Mn²⁺) ions along the a-direction (Figure 6b). For convenience, these chains will be referred to as J₂- and J₁- chains, respectively. Note that each J₂-chain is coupled to two adjacent J₂-chains and also to two adjacent J₁-chains. Between adjacent J₁- and J₂-chains four different spin exchange paths (i.e., J₄, J₅, J₆ and J₇) occur (Figure 6c), leading to (J₁, J₄, J₅), (J₂, J₄, J₅), (J₁, J₆, J₇) and (J₂, J₆, J₇) exchange triangles. A more extended view of Figure 6c is presented in Figure S1 in the Supporting Information (SI).

To determine the values of these exchanges, we employ the spin Hamiltonian defined as

$${\mathrm{H}}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=-{\sum }_{\mathrm{i}>\mathrm{j}}{J}_{ij}{\overrightarrow{\mathrm{S}}}_{\mathrm{i}}·{\overrightarrow{\mathrm{S}}}_{\mathrm{j}},$$

(2)

where the spin exchange J_ij between two spin sites can be any one of J₁–J₇. To evaluate J₁–J₇, we carry out the energy-mapping analysis [2,3,4] using the eight ordered spin states, i.e., AFi, where i = 1–8, depicted in Figure S2 of the SI. First, we express the energies of the eight ordered states in terms of the spin exchanges J₁–J₇ using the spin Hamiltonian of Equation (2) and then determine the relative energies of these states (

Table 1. Relative energies (in meV/FU) of the eight ordered spin states obtained from DFT + U calculations.

Ordered Spin States	Ueff = 3 eV	Ueff = 4 eV
AF1	10.01	8.27
AF2	8.80	7.20
AF3	6.40	5.32
AF4	8.49	7.81
AF5	17.84	15.03
AF6	13.26	12.68
AF7	3.62	3.29
AF8	0	0

) by DFT calculations using the frozen core projector augmented plane wave [16,17] encoded in the Vienna ab Initio Simulation Package [18] and the exchange-correlation functional of Perdew, Burke and Ernzerhof [19].

The electron correlations associated with the 3d states of Mn were taken into consideration by DFT + U calculations with effective on-site repulsion U_eff = U − J = 3 eV and 4 eV [20]. All our DFT + U calculations used the plane wave cutoff energy of 450 eV, a set of (6 × 4 × 4) k-points, and the threshold of 10⁻⁶ eV for self-consistent-field energy convergence. Finally, the numerical values of J₁–J₇ were obtained by mapping the relative energies of the eight ordered spin states onto the corresponding energies determined by DFT + U calculations. The results of these energy-mapping analyses are summarized in

Table 2. Geometrical parameters of the exchange paths and the values of the spin exchanges in Ba₂Mn₃(SeO₃)₆. The plus and minus signs of the spin exchanges represent ferromagnetic and antiferromagnetic couplings, respectively.

Path	Geometrical Parameters		Spin Exchanges (in K)
	Ions Involved	Distance, Å	Ueff = 3 eV	Ueff = 4 eV
J1	Mn1…Mn1	5.4717	−2.23	−1.74
J2	Mn2…Mn2	5.4717	−2.62	−2.14
J3	Mn2…Mn2	5.4655	1.51	1.80
J4	Mn1…Mn2	5.9554	−2.38	−1.63
J5	Mn1…Mn2	6.0883	−0.88	−0.45
J6	Mn1…Mn2	5.9737	−2.05	−1.43
J7	Mn1…Mn2	6.1063	3.88	3.56

.

As already pointed out, each J₂-chain interacts with two adjacent J₂-chains and with two adjacent J₁-chains (Figure 6b). In terms of the spin exchanges of Table 2, the nature of these interchain interactions can be stated as follows:

(1) Each J₁-chain is an antiferromagnetic chain, and so is each J₂-chain.

(2) Each J₂-chain is ferromagnetically coupled to two adjacent J₂-chains via the exchange J₃ (Figure 7a).

(3) Each J₂-chain is coupled to one J₁-chain via the antiferromagnetic exchange J₆ and the ferromagnetic exchange J₇ (Figure 7b), forming the (J₁, J₆, J₇) and (J₂, J₆, J₇) exchange triangles. With one ferromagnetic and two antiferromagnetic exchanges, each exchange triangle is not spin-frustrated, so the coupling between these J₂- and J₁-chains is ferromagnetic.

(4) Each J₂-chain is coupled to another J₁-chain via the antiferromagnetic exchanges J₄ and J₅ (Figure 7c), forming the (J₁, J₄, J₅) and (J₂, J₄, J₅) exchange triangles. With all three antiferromagnetic spin exchanges, each exchange triangle is spin frustrated. Thus, as depicted in Figure 7c,d, one can have two different spin arrangements between these J₁- and J₂-chains. This explains the presence of spin frustration in Ba₂Mn₃(SeO₃)₆ as indicated by its index of spin frustration of f = 5. Since J₄ is more strongly antiferromagnetic than J₅ (by a factor of approximately 3), the spin configuration of Figure 7c is energetically more stable than that of Figure 7d. The antiferromagnetic ordering at T_N = 5.7 K means that the spin configuration of Figure 7c dominates over that of Figure 7d in the population.

(5) As already noted, each J₂-chain is an antiferromagnetic chain and interacts with two adjacent J₂-chains and two adjacent J₁-chains. These interchain interactions are all ferromagnetic except for the one with one of the two J₁-chains. The latter is spin-frustrated as described above. Above T_N = 5.7 K, where the latter spin frustration is not settled, the magnetic behavior of Ba₂Mn₃(SeO₃)₆ should have a strong 1D antiferromagnetic chain character because the antiferromagnetic J₁- and J₂-chains are ferromagnetically coupled (via ferromagnetic J₇ and antiferromagnetic J₆, Figure 6b). This explains the occurrence of the broad maximum in the magnetic susceptibility and the specific heat of Ba₂Mn₃(SeO₃)₆. However, interactions between the J₁- and J₂-chains via J₄ and J₅ are spin-frustrated, because the (J₁, J₄, J₅) and (J₂, J₄, J₅) exchange triangles are spin-frustrated so two different arrangements between the J₁- and J₂-chains are possible (Figure 6c,d).

The two sets of the spin exchanges obtained with U_eff = 3 and 4 eV are similar in trend. To see which set is better, one might estimate the Weiss temperature Θ using the mean field theory [21,22], to see which set leads to a value closer to the experimental value of Θ = −27.8 K observed for Ba₂Mn₃(SeO₃)₆. According to Figure 6a,c, the Θ_calc value for Mn²⁺ (S = 5/2) ions is predicted to be

$${\Theta }_{calc}=\frac{S\left(S+1\right)}{3}(2J_1+2J_4+2J_5+2J_6+2J_7),$$

(3)

which is –21.9 K for U_eff = 3 eV and –9.9 K for U_eff = 4 eV. Similarly, the Θ_calc value for Mn³⁺ (S = 2) ions is predicted to be

$${\Theta }_{calc}=\frac{S\left(S+1\right)}{3}(2J_2+2J_3+2J_4+2J_5+2J_6+2J_7)$$

(4)

which is −10.2 K for U_eff = 3 eV and −1.2 K for U_eff = 4 eV. Thus, the spin exchanges obtained from of U_eff = 3 eV better describes the observed Weiss temperature.

4. Discussion and Conclusions

In summary, our magnetization and specific heat measurements of Ba₂Mn₃(SeO₃)₆ reveal that it is a low-dimensional antiferromagnet with a short-range one-dimensional antiferromagnetic chain behavior followed by a long-range antiferromagnetic order as marked by a succession of a broad maximum at T_corr = 10.1 ± 0.1 K and a sharper anomaly at T_N = 5.7 ± 0.1 K in both magnetic susceptibility (Fisher’s specific heat) and specific heat. These observations are well-explained in terms of the spin exchanges of Ba₂Mn₃(SeO₃)₆, both antiferromagnetic and ferromagnetic, evaluated by the energy-mapping analysis. In the ordered state, Ba₂Mn₃(SeO₃)₆ exhibits a metamagnetic phase transition inherent for the Ising magnets with magnetization hysteresis most pronounced at low temperatures. Notably, no hysteresis is observed at µ₀H = 0 which points to the absence of spontaneous magnetization in Ba₂Mn₃(SeO₃)₆.

Structurally, the title compound is organized by the honeycomb layers of Mn³⁺ ions alternating with the triangular layers of Mn²⁺ ions. Magnetically, it consists of uniform chains of integer spins S = 2 of Mn³⁺ ions and half-integer spins S = 5/2 of Mn²⁺ ions running along the a axis. Qualitatively different quantum ground states, i.e., gapped and gapless spin liquid, correspondingly, can be expected for these entities [23]. However, the interaction of these chains leads to the formation of a long-range antiferromagnetic order. It would be interesting to synthesize and study isostructural phases of Ba₂Mn₃(SeO₃)₆ where either the divalent or the trivalent magnetic ions is replaced with nonmagnetic counterparts. The exchange interactions through a chalcogenide anion such as SeO₃^2- makes the scales of magnetic fields and temperatures quite convenient for experiments with equipment readily available.

While preparing this article, we became aware of an independent unpublished study on Ba₂Mn₃(SeO₃)₆ [24], which reported experimental data similar to ours but did not provide any theoretical analysis.