1. Introduction

The orthorhombic semiconductor Ag₂FeGeSe₄ which crystallizes with the wurtz-stannite structure ^1,2 belongs to family of the I₂-II-IV-VI₄ compounds where I = Cu or Ag;II = Zn, Cd, Hg, Mn, Fe or Co; IV = Si, Ge, Sn or Pb; and VI = S, Se or Te. These materials are of great interest because of both their applications in the fabrication of low cost solar cells ³ and their large magneto-optical effects which are observed when II atoms are paramagnetic ions ^4,5. The study of the magnetization of Ag₂FeGeSe₄ over the temperature range from 2 to 300 K shows that this material has an antiferromagnetic behavior ⁶. The large numbers of known antiferromagnetic substances exhibit a rich variety of magnetic phase transitions. These transitions are described with particular emphasis in uniaxial antiferromagnets of the easy -axis type. Susceptibility measurements made at high magnetic field (5 T) showed a magnetic moment of 4.4 μB for ion Fe⁺² in acceptable agreement with the theoretical value of 4.9 μB obtained when J = S. Measurements of magnetization at high field pulsed technique up to 32 T have also been made on this compound ⁶.

The present paper describes the behavior of the critical-fields and the thermal broadening for Ag₂FeGeSe₄ at various temperatures obtained in a previous work ⁶. To our knowledge, there are no previous reports on thermal broadening in Ag₂FeGeSe₄ compound, and the literature information on it is very scarce.

2. Theoretical details

Theoretical and experimental analysis e.g.^4,7,8,9 for an antiferromagnetic material show that three phases can occur in the magnetic field-temperature (B - T) plane of the magnetic phase diagram. These are the paramagnetic (P), the antiferromagnetic (AF) and the spin-flop (SF). At the AF to SF first-order phase transition the magnetization increases linearly with applied field and show a discontinuity, whereas the SF to P phase transition is of second-order type and the magnetization breaks off from this line ¹⁰. These analyzes have been concerned mainly with uniaxial materials with an easy-axis z. It is seen in such a case that the details of the diagram depend on the orientation of the applied magnetic field with the z direction. Critical fields for these transitions are obtained as a function of temperature and the phase diagram in the B - T plane can thus be determined.

According to the scheme proposed by Ehrenberg et al.⁷, for sufficiently low-fields in the AF region, the magnetization is proportional to the applied magnetic field as given by the expression,

where μ0 is the permeability of free space and χ is the magnetic susceptibility of the AF ground state often tabulated in terms of the mass susceptibility χg=χ/ρ. Here ρ is the mass density which is 6.83 × 10³ kg/m³ for Ag₂FeGeSe₄ as calculated from the lattice parameters reported in Ref. ².

On the other hand, for sufficiently high fields, the magnetic moments are forced into a ferromagnetic-like arrangement, which belongs to the P region in the phase diagram, where all the magnetic moments are aligned with the magnetic field. In this paramagnetic region, the expressions usually applied to single crystals for describing the field and temperature dependence of the magnetization, based on a quantum mechanical description of the phenomena through the function of Brillouin are expected to be inappropriate for polycrystalline samples. This is because polycrystals are macroscopic objects that combine thousands of spins at random. The simplest approximation in such a case is an isotropic distribution of easy axes at angle θ from the applied field B. Therefore, a classical explanation via the Langevin function is justified. Thus, the field and temperature dependence of magnetization in this phase can be expressed as

where M _S is the saturation magnetization and L(x) is the Langevin function which is given by

with x=μBkB/T, μ is the magnetic moment, k _B is the Boltzmann constant and T is the absolute temperature.

Finally, in the SF phase, the magnetization is proportional to the applied field strength corrected in real systems by an offset M ₀:

For all magnetizations curves below of the Néel temperature the field dependence of the magnetization, assuming the existence of an SF phase, can be described by the following expression:

Here B _SF and B _P are the critical field strengths for the transitions AF→SF and SF→P, σSF and σP are a measure for the width of the thermal broadening of these transitions.

3. Results and Discussion

The magnetization of Ag₂FeGeSe₄ at 2 K as a function of the applied magnetic field is shown in Fig. 1a). Only the up-stroke cycle is presented in this figure. The application of the external field induces the two successive AF→SF and SF→P phase transitions expected for this compound. These are observed at about 16 and 25 T, respectively. The stability ranges of these phases are summarized in the phase diagram of Fig. 1b).

Figure 1. (a) Magnetization M against external applied magnetic field B at 2 K from pulsed field measurements for upstroke cycle. The dashed vertical lines show the positions of the B _SF and B _P points (see Table I). The solid yellow line is a fit of the data (open circles) by using Eq. (5) extrapolated until 35 T. The contributions from the three terms: M _AF (dash blue line), M_SF (dot red line) and M_P (dash-dot green line) are also shown. (b) The magnetic phase diagram for the Ag₂FeGeSe₄ compound showing the experimental values for field-up (full points) and the calculated values (open circles) for up-stroke cycle from Table I. The solid lines are guides for eyes extrapolated until the Néel point. Adapted from Ref. ⁶. 

The solid yellow line in Fig. 1a) represents a fit of Eq. (5) to the experimental M (B) data. It is clear from this figure that the saturation magnetization was not reached in the range of magnetic field values achieved in this work. Also, an appreciable hysteresis effect was observed for all curves (see Fig. 2 in Ref. ⁶). For this reason, the M _S values were fixed during the fitting for all the curves in the down-stroke cycle, not shown here, at the values obtained in the up-stroke cycle.

Figure 2. The solid line represents the best fits with Eq. (5). The open circles are 2 K data on polycrystalline a) Ba₃Cu₃In₄O₁₂ and b) Ba₃Cu₃Sc₄O₁₂ taken from Ref. ¹⁴. 

Table I summarizes the results obtained by fitting with Eq. (5) the curves of magnetization at various temperatures for up-stroke cycles. The susceptibilities calculated and the critical field strength B _SF agree with the experimental values previous report in Ref. ⁶. It is worth mentioning that attempt was also made to adjust the magnetizations curves by using the Brillouin function. However, no fit was obtained. This is probably due to the polycrystalline nature of the present sample as was explained above.

Table I. Parameters obtained from fitting Eq. (5) to curves M vs. B at various temperatures. The BP parameter was taken from the Ref. ⁶ and the μ parameter was fixed at μexp. = 4.90 (Fe⁺²). Numbers in parentheses are statistical errors. For B _P values was estimated an error of ±2 T. 

Note. The labels U and D mean up-stroke and down-stroke cycles, respectively.

For a uniaxial antiferromagnet of the easy-axis type, at temperatures below the Neel temperature T _N , the orientation of M _l and M ₂ relative to the crystallographic axes is determined by the anisotropy energy. As indicated by Shapira ¹¹, The AF→SF transition occurs at

where K is the effective anisotropy energy per unit mass, α is the susceptibility in the SF phase and χ is the susceptibility in the P phase. It equation can be rewrite as,

Thus, the values for K at various temperatures were calculated from Eq. (7) (see Table I). For above of 10 K, the values for susceptibility α in the SF phase are higher that the susceptibility χ in the AF phase, as expected. However, for 2 and 4 K occurs the contrary. It means that at these temperatures the thermodynamic potential (from Eq. (6) in Ref. ¹¹) in the AF phase is higher than the thermodynamic potential in the SF phase. It is usually found that magnetic materials that have a larger symmetry leads to lower anisotropy, so materials with cubic crystal structure have lower anisotropy energy than hexagonal materials. Thus, the effective anisotropy energy of BaFe₁₂O₁₉ with hexagonal structure is 4.7 × 10⁶ erg/cm³¹². In our case, for Ag₂FeGeSe₄ with a pseudo-cubic structure and using the density value (6.83 × 10³ kg/m³), we have obtained anisotropy energy values of order 10³ erg/cm³, i.e., of same order that Ni (cubic structure) ¹³.

Since for Ag₂FeGeSe₄ the saturation of the magnetization was not reached experimentally, in order to test the present model in the region of high fields where the magnetization saturates, M(B) curves for Ba₃Cu₃In₄O₁₂ and Ba₃Cu₃Sc₄O₁₂ compounds recently reported ¹⁴ were fitted using Eq. (5). The obtained parameters are summarized in Table II. It can be seen that the values for B _P and B _SF obtained for these compounds agree quite well with those reported. Also, their widths σSF and σP are reasonable physically. As expected, the thermal broadening for the transition SF→P is greater than that for the transition AF→SF. These results confirm the validity of the present model. Kumar et al. ¹⁵ used a microscopic 1D model of the P phase combined with a phenomenological model based on sublattice magnetization for crystals with SF or AF transitions that describes the magnetization curve and that only provides information on critical field strengths of the transitions.

Table II. Parameters obtained for Ba₃Cu₃In₄O₁₂ and Ba₃Cu₃Sc₄O₁₂ compounds with Eq. (5). Numbers in parentheses are statistical errors. The μ parameters for Cu²⁺ were taken from Ref. ¹⁴. 

^*This work.

In summary, we have shown that the scheme proposed by Ehrenberg et al.⁷ to study the temperature and magnetic field dependence of magnetization in the AF, SF and P phases of the low anisotropy antiferromagnetic materials, success describe all magnetization curves below Néel temperature for the Ag₂FeGeSe₄ compound. In the present case, for the analysis of the paramagnetic region, the model was adapted for polycrystalline samples by using the classical Langevin function instead of the Brillouin one normally used for single crystals. Transitions given by B _SF and B _P , lead to the magnitudes χ, α, σSF, σP and K in Table I. The parameters χ, α, σSF, σP and K provide a consistent picture and a starting point for more detailed studies of the magnetic interactions in this type of compounds.