PACS: 75.50.Pp; 75.50.Ee; 75.60.Ej

1. Introduction

The orthorhombic semiconductor Ag_{2}FeGeSe_{4} which crystallizes with the wurtz-stannite structure ^{1}^{,}^{2} belongs to family of the I_{2}-II-IV-VI_{4} 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 Ag_{2}FeGeSe_{4} 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 ^{+2} in acceptable agreement with the theoretical value of 4.9 *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 Ag_{2}FeGeSe_{4} at various temperatures obtained in a previous work ^{6}. To our knowledge, there are no previous reports on thermal broadening in Ag_{2}FeGeSe_{4} 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 ^{3} kg/m^{3} for Ag_{2}FeGeSe_{4} 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 *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 *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

3. Results and Discussion

The magnetization of Ag_{2}FeGeSe_{4} 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

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.

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.

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

where *K* is the effective anisotropy energy per unit mass,

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 ^{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 BaFe_{12}O_{19} with hexagonal structure is 4.7 × 10^{6} erg/cm^{3}^{12}. In our case, for Ag_{2}FeGeSe_{4} with a pseudo-cubic structure and using the density value (6.83 × 10^{3} kg/m^{3}), we have obtained anisotropy energy values of order 10^{3} erg/cm^{3}, *i.e.*, of same order that Ni (cubic structure) ^{13}.

Since for Ag_{2}FeGeSe_{4} 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 Ba_{3}Cu_{3}In_{4}O_{12} and Ba_{3}Cu_{3}Sc_{4}O_{12} 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 *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.

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 Ag_{2}FeGeSe_{4} 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 *K* in Table I. The parameters *K* provide a consistent picture and a starting point for more detailed studies of the magnetic interactions in this type of compounds.