## 1 Introduction  
The magnetic and transport properties of AB\({}_{2}\)X\({}_{4}\) chalcogenide spinels are strongly influenced by the distribution of the metal ions in the structure. Because the cationic and anionic distribution can change when substitution occurs, the physical properties of the spinel material may differ significantly from those of the parent spinels. An interesting property of spinels containing Cr is magnetoresistance, which refers to the change in the electrical resistance of materials in the presence of a magnetic field [1]. This has been the focus of many studies because of its potential for use in new data storage devices with reduced energy consumption [2, 3]. Therefore, solid solutions of seleno- and thiospinels have been well studied because of their extraordinary electrical and magnetic properties, which can differ significantly by substitution [4, 5, 6, 7, 8, 9, 10, 11]. For example, replacing metal ions in the octahedral or tetrahedral position with diamagnetic ions can alter the magnetic properties of CuCr\({}_{2}\)S\({}_{4}\) and CuCr\({}_{2}\)Se\({}_{4}\). In (Cu)\({}_{\rm tet}\)[Cr\({}_{2}\)\({}_{x}\)M\({}_{x}\)]\({}_{\rm oct}\)Se\({}_{4}\) spinels, the substitutions of Cr by M (M = Zr, Sn) have indicated that M is present as a 4\({}^{+}\) ion, and Cu as a diamagnetic Cu\({}^{+}\) ion [3, 12]. For CuCr\({}_{2}\)S\({}_{4}\), its physical properties can be understood if its charge distribution is assumed to be Cu\({}^{+}\)Cr\({}^{3+}\)Cr\({}^{4+}\)S\({}^{8-}\); thus, metallic conduction and ferromagnetism are ascribed to the double exchange between Cr\({}^{3+}\) and Cr\({}^{4+}\)[13, 14]. Looking at the magnetic phase diagram of (Cu)\({}_{\rm tet}\)[Cr\({}_{2}\)\({}_{x}\)T\({}_{\rm x}\)]\({}_{\rm oct}\)S\({}_{4}\)[15], which results from the substitution of Cr by Ti, the magnetic frustration generated changes the ferromagnetic properties to a spin-glassregime. Regarding (Cu)\({}_{\rm{tet}}\)[Cr\({}_{2.x}\)Sn\({}_{x}\)]\({}_{\rm{oct}}\)Se\({}_{4}\), where Sn replaced Cr, ferromagnetic interaction at x = 0.3 was noticeable, and for x = 0.7, ferromagnetic character was predominantly present in the antiferromagnetic (AF) behavior [6]. The degree of Se, Cu, or Cr substitution can modify the magnetic and electrical properties of chalcogenipels. For example, CuCr\({}_{2.x}\)Sn\({}_{x}\)S\({}_{2}\)Se\({}_{2}\) samples with \(x\) = 0.2 and 0.4 showed ferromagnetic interaction, whereas high transition temperature (AF) behavior occurred in \(x\) = 0.6, 0.8, and 1.0 samples [13]. In another case, the electrical conduction dependence on Cu\({}_{1.x}\)Co\({}_{x}\)Cr\({}_{2}\)Se\({}_{4}\) exhibited metallic-type conductivity within the chemical 0.1 \(\leq x\leq\) 0.5 range [16; 17]. The substitution of Li by Cu in the CuCr\({}_{2}\)Se\({}_{4}\) end-member caused a change from metallic to insulating behavior, which was ascribed to the suppression of holes in the ligand non-bonding \(p\) orbitals of the selenium atoms resulting from the substitution [18]. Similarly, changes occurred when chalcogen was substituted, as in CuIr\({}_{2}\)(S\({}_{1.y}\)Se\({}_{y}\))\({}_{4}\), with \(y\) \(\leq\) 0.15, as metal-insulated transition occurred along with a structural modification from a cubic to tetragonal structure. In CuCr\({}_{2}\)S\({}_{4.x}\)Se\({}_{x}\), semiconduction appeared when CuCr\({}_{2}\)S\({}_{4}\) and CuCr\({}_{2}\)Se\({}_{4}\) spinels were mixed with metallic conduction [19; 20]. [PERSON] et al. reported the magnetoresistance of CuCrZrS\({}_{4}\). This solid solution exhibited variable range hopping (VHR) behavior and decreasing resistivity with increasing magnetic field [21]. For other compounds, there was a progressive change in the electrical properties. Co\({}_{0.1}\)Fe\({}_{0.9}\)Cr\({}_{2}\)S\({}_{4}\) showed a temperature dependence from metallic behavior (140-178 K) to semiconducting behavior at \(T\) > 178 K [22]. In addition, this material showed a linear decrease in electrical resistance as the applied external magnetic field increased.