First-principle investigation of the electronic and magnetic properties of PbMn(SO{}_{\text{4}}){}_{\text{2}}

Abstract

The magnetic properties of oxide PbMn(SO₄)₂ consisted of MnO₆ octahedra which connected with each other through SO₄ tetrahedra, are well studied in experiments. In this paper, we explored its interesting electronic and magnetic properties with first-principle calculations. Our results show that all Mn ions have high spin states, namely, S=5/2, and the magnetic couplings between NN and NNN are antiferromagnetic, which agree well with the experimental results. Besides, the surprising results of spin exchange interactions between the NN and NNN are excellently explained with extended Hückel tight-binding calculations.

Graphical abstract

Keywords

first-principle calculations / magnetic properties / spin exchange

1 Introduction

The crystal structure of the magnetic oxide PbMn(SO₄)₂ was first studied by West et al. [1]. As illustrated in the Fig. 1, the distorted octahedra MnO₆ are connected with each other through the SO₄ tetrahedra. The chain formed by the Mn²⁺ ions along a direction alternate with the chain of the Pb²⁺ ions along b direction. Besides the crystal structure, West et al. also reported the magnetic properties of PbMn(SO₄)₂ by using the powder x-ray and neutron diffraction [1]. According to their investigation, PbMn(SO₄)₂ undergoes antiferromagnetic (AFM) three-dimensional (3D) order below \sim 5.5 K (i.e., T_N\approx 5.5 K). The Curie–Weiss temperature θ of PbMn(SO₄)₂, reported to be -10.7 K, shows that dominant AFM interactions exist. Since T_N is considerably smaller than |\theta |(i.e., |\theta |/T_N\approx 2), substantial geometric spin frustration [2, 3] should be present in PbMn(SO₄)₂.

After checking the atomic environment, we found that each Mn²⁺ ion has four nearest neighbors (NN) Mn²⁺ ions, and they are connected by the double Mn–O–S–O–Mn path ways. On the other hand, each Mn²⁺ ion is also surrounded by four next nearest neighbors (NNN) Mn²⁺ ions and connected by a single Mn–O–S–O–Mn path way. However, the nature and strength of the spin exchange interactions between the NN and NNN of Mn²⁺ ions have not been determined so far. To our knowledge, the strengths of J₁, J₂ between the NN and NNN are the super-super exchange (SSE) interactions and proportional to Curie–Weiss temperature θ. The relationship between the Curie–Weiss temperature θ and the spin exchanges can be described as: \theta =\frac{S(S+1)}{3k_{\mathrm{B}}}{\sum }_i{z}_i{J}_i [4], where the summation runs over all nearest neighbors of a given spin site, z_i is the number of nearest neighbors connected by the spin exchange parameter J_i, and S is the spin quantum number of each spin site (i.e., S=5/2 in the present case). In terms of the experimental Curie–Weiss temperature θ (-10.7 K), we can predict that the strength of SSE J₁ and J₂ should be very weak, and the stronger one should be AFM.

In order to investigate the novel properties of PbMn(SO₄)₂, we theoretically study the electronic and magnetic properties of PbMn(SO₄)₂ for the first time through electronic calculations and evaluating the spin exchange interactions by employing the mapping analysis based on density functional electronic structure calculations [5-12]. To extract the values of J₁, J₂ on the basis of density functional calculations, three ordered spin states of PbMn(SO₄)₂ are adopted, as depicted in Fig. 2.

2 Computational method

Our spin-polarized density functional calculations employed the projector augmented wave method implemented in Vienna ab initio simulation package [13-15] with the generalized gradient approximation (GGA) [16] for the exchange-correlation functional, the plane-wave cutoff energy of 400 eV, a set of 5\times 2\times 2 points [17], and the threshold 10^-5eV for energy convergence. In general, the electron correlation associated with 3d states is not well described by density functional calculations, and this deficiency is commonly corrected by using the GGA plus on-site repulsion U (GGA+U) method. Considering the electron correlation of Mn 3d, we carried out GGA+U calculations with U=3.0 eV [18, 19].

3 Results and discussion

Figure 3 shows plots of the total density of states (DOS) and the projected DOS (PDOS) for the Mn 3d states, obtained from GGA+U (U=3.0 eV) calculations for the ferromagnetic (FM) state of PbMn(SO₄)₂. As we can see, the band gap is predicted to be about 2.5 eV by the GGA+U calculations. All up-spin Mn 3d states are occupied and are separated with a band gap from the all down-spin states lying above. This is consistent with the finding that PbMn(SO₄)₂ consists of high-spin Mn²⁺ (d⁵) ions and is a magnetic insulator.

For the magnetic properties of PbMn(SO₄)₂, the accurate spin exchange parameters are needed. To determine the values of J₁, J₂, we express the energies of the three ordered spin states in terms of the spin Hamiltonian,

in which J_{ij}(=J_1, J₂) is the exchange parameter for the interaction between the spin sites i and j, while \widehat{S_i} and \widehat{S_j} are the spin angular momentum operators at the spin sites i and j, respectively. By applying the energy expressions obtained for spin dimers with N unpaired spins per spin site (in the present case, N=5 [20, 21]), the total spin exchange energies (per formula unit) of the three ordered spin states are written as:

\begin{array}{}\text{(2)}& E_{\mathrm{FM}}=(-2J_1-2J_2){\displaystyle \frac{N^2}{4}}\end{array}

\begin{array}{}\text{(3)}& E_{\mathrm{AF1}}=(+J_1){\displaystyle \frac{N^2}{4}}\end{array}

\begin{array}{}\text{(4)}& E_{\mathrm{AF2}}=(-J_2){\displaystyle \frac{N^2}{4}}\end{array}

The relative energies of the three ordered spin states of PbMn(SO₄)₂ (Fig. 2), obtained from our GGA+U (with U=3.0 eV), are summarized Table 1. Therefore, the values of J₁–J₂ listed in Table 2 are determined by equating the relative energies of the three ordered spin states obtained from GGA+U calculations to the corresponding relative energies obtained from the total spin exchange energies.

From the spin exchange values of Table 2, the following are observed: (i) As we expect in the introduction, the spin interactions J₁, J₂ between the Mn²⁺ ions are very weak because of the low Neél temperature. (ii) In the GGA+U calculations, both of J₁ and J₂ spin interactions are AFM exchanges, and the stronger one is J₂. As we know, antiferromagnetic exchanges are inversely proportional to the electronic correction U value, which means larger U will reduce the spin interactions. But the general picture that J₂ is stronger than J₁ will not be changed by the U value. In our studies, we have adopted U value as 3 eV, which is reasonable for 3d transition metal atoms, and the results agree well with the observed experimental data. Thus, our results should be reliable. (iii) In the GGA+U calculations, the AFM exchanges J₁, J₂, J₁ triangles lead to spin frustration.

It is well known that the local octahedral distortions have important consequences on the nature and strength of the spin exchanges. Therefore, we present geometrical parameters associated with the spin exchange paths J₁–J₂ in Table 3. As we can see, the <M–O–O and <O–O–M angles of the SSE paths J₁ and J₂ are greater than 90. Therefore, the magnetic orbitals of Mn ions can overlap with each other, resulting in the AFM coupling [22]. Both of SSE J₁ and J₂ are involving Mn–O\mathrm{\cdots }O–Mn paths in PbMn(SO₄)₂, and J₁ has a shorter Mn–Mn distance than J₂ (4.943 Å vs. 6.733 Å). Nevertheless, the strength of an SSE interaction M–O\mathrm{\cdots }O–M, where M is a magnetic transition-metal ion, is essentially determined by the overlapping between the O orbtials. In general, the O orbital overlapping depends on the O\mathrm{\cdots }O contact distance and is expected to become strong when the O\mathrm{\cdots }O contact distance is short. Table 3 indicates that J₂ has a slightly shorter O\mathrm{\cdots }O contact distance than does the J₁ (2.377 Å vs. 2.435 Å or 2.407 Å). It is not surprising that the strength of J₂ is larger than J₁, but unbelievable that J₂ is more than eight times of J₁.

To get more clear pictures about the magnetic coupling, we use the extended Hückel tight-binding calculations to analyze the calculated spin exchange parameters. In this part, we will explain why the super-superexchange (SSE) J₂ is much stronger than J₁. To explain this apparently surprising result, we determine the magnetic orbitals of the Mn₂O₁₆S₂ and Mn₂O₁₄S involving the J₁ and J₂ SSE paths by extended Hückel tight-binding calcaulations. As presented in Fig. 4, magnetic orbitals of Mn₂O₁₂ are composed of the anti-bonding Mn 3d and O 2p orbtials. The overlapping of O 2p orbitals involving the spin exchange path J₁ is almost zero because the O 2p orbitals are perpendicular to each other, while the overlapping of O 2p orbitals becomes much stronger because of the parallel O 2p orbitals involving the spin exchange path J₂. Generally speaking, AFM spin exchange J depends on the overlap of magnetic orbitals, and J_{\mathrm{AF}} becomes negligible if the overlap integral between the magnetic orbitals is small. Therefore, the spin exchange J₂ is strongly AFM because of the large overlapping, other than the geometrical distance.

4 Conclusion

We have performed comprehensive first-principle calculations to investigate the electronic and magnetic properties of PbMn(SO₄)₂. Our results show that all Mn ions have high spin states, namely, S=5/2. The magnetic properties are studied based on the spin exchanges J₁–J₂ of PbMn(SO₄)₂ extracted from first-principle calculations. Both J₁ and J₂ are AFM, and J₂ is much stronger than J₁, and J₁, J₂, J₁ triangles lead to spin frustration which is consistent with the experiment result. Besides, the intrinsic mechanic leading to the stronger J₂ is well explored with extended Hückel tight-binding calcaulations.

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