Effect of magnetic dilution in Zn1−xMnxGa2Se4 „0<x<0.5..

The profound effect of magnetic dilution in Zn1−xMnxGa2Se4 with 0 x 0. 5 is demonstrated by the analysis of neutron powder diffraction and magnetic susceptibility data. All the members of the series show a defective stannite crystal structure and a random distribution of the Mn ions. The magnetic archetype system of Zn1−xMnxGa2Se4 with 0 x 0. 5 is not found to be MnGa2Se4 the magnetic end member of the series with I4̄ symmetry but, instead, a hypothetical compound with I4̄2m symmetry. The distance between the Mn ions along the various magnetic superexchange pathways is not seriously affected by the degree of dilution. However, the global magnetic interaction between manganese ions is found to be significantly higher than that expected for the classical magnetic dilution of the parent MnGa2Se4. A generally applicable method for the calculation of the correct Weiss temperature in diluted magnetic semiconductors is provided. © 2005 American Institute of Physics. DOI: 10.1063/1.1944220


I. INTRODUCTION
The combination of two or more apparently independent areas of research often leads to valuable observations. This is the case for diluted magnetic semiconductors ͑DMSs͒. Within these materials a given concentration of a magnetic element is introduced into a nonmagnetic semiconductor matrix. The interaction between the conduction and/or valence band electrons and the localized magnetic moments of the magnetic ions gives rise to an unusual combination of magnetic, electronic, and optical properties that make DMSs important subjects for both scientific investigation and potential device applications. 1,2 Specific properties and requirements are matched by tailoring the composition through the use of solid solutions, 3,4 though significant differences between the crystal structure of magnetic materials and a nonmagnetic host can make the two systems incompatible with each another. 5 The degree of atomic ordering critically influences the physical properties ͑optical, electronic, magnetic,…͒ of the material. 6,7 A precise knowledge of the distribution of the cations in the crystal lattice is, therefore, of significant importance. In most DMSs the arrangement of the magnetic atoms is random. 8 As a consequence, many variables follow a linear dependence versus the content of the magnetic cations, as is the case of the Weiss temperature. 9 One of the problems encountered in DMSs is that most of these systems, particularly pseudobinary DMSs, exhibit comparatively higher values. 10 This makes the calculation of reliable Weiss temperatures more difficult due to experimental limitations on the available temperature range.
Mn 2+ is a magnetic ion with spin S =5/2. ZnGa 2 Se 4 and MnGa 2 Se 4 are semiconductors showing direct gaps of around 2.3 and 2.5 eV, respectively. 11,12 The controlled substitution of Zn ions by Mn in ZnGa 2 Se 4 generates the series of DMSs Zn 1−x Mn x Ga 2 Se 4 . Since the crystal structure of both end members of the family are rather similar ͑defective stannite for ZnGa 2 Se 4 and defective chalcopyrite for MnGa 2 Se 4 ͒, a high degree of miscibility is expected. Moreover, MnGa 2 Se 4 exhibits a totally ordered scheme of the magnetic cations while ZnGa 2 Se 4 shows a completely disordered arrangement of the Zn atoms. 13,14 Interesting behavior is then expected within this series due to the competition of both order and disorder trends. From the magnetic point of view, relatively low Weiss temperature values can be anticipated since MnGa 2 Se 4 orders magnetically below 6.4± 0.1 K. 15 The aim of this research is to explore the influence of the distribution of the Mn ions on the magnetic behavior of Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0. 5, which is shown to differ significantly from that expected for the classical dilution of a magnetic semiconductor. Thus, we analyze the dependence of the magnetic superexchange constants with magnetic dilution and the magnetic archetype system corresponding to this series. The combination of neutron powder diffraction experiments and magnetic susceptibility measurements is shown to be a useful tool for the determination of correct Weiss temperatures in DMSs.

II. EXPERIMENT
Orange single crystals of Zn 1−x Mn x Ga 2 Se 4 with x = 0.00, 0.104͑4͒, 0.240͑4͒, 0.343͑4͒, and 0.482͑4͒ were synthesized using the chemical vapor transport method. 16 In order to determine the distribution of magnetic ions in the samples under study, neutron diffraction experiments are considered more suitable than the more common x-ray technique. Mn is more clearly differentiated from other elements using neutrons due to its negative coherent nuclear scattering length. Room-temperature time-of-flight neutron-diffraction experiments were performed on the Polaris powder diffractometer 18 at the ISIS source, UK. Thin-walled vanadium cans acted as sample containers. Diffraction data were collected in all available detector banks over a d-spacing range 0.5Ͻ d͑Å͒ Ͻ 12.0. Normalization of the data and correction for sample absorption were carried out using the GE-NIE software. 19 The program TF14LS, based on the CAM-BRIDGE CRYSTALLOGRAPHIC SUBROUTINE LIBRARY, 20 was used for Rietveld analysis of the diffraction patterns. Refinements of the crystal structure were carried out using the data in the range 0.8Ͻ d͑Å͒ Ͻ 3.2 obtained using the highestresolution backscattering detectors, which possess an essentially constant resolution of ⌬d / d Ϸ 5 ϫ 10 −3 .
DC magnetic susceptibility measurements were performed on a magnetometer with superconducting quantum interference device ͑SQUID͒ detection manufactured by Quantum Design. The data were collected using polycrystalline samples under an external magnetic field of 1 T. The temperature was varied between 5 and 300 K.

A. Structure
The crystal structures of both ZnGa 2 Se 4 and MnGa 2 Se 4 were used as the starting models for the determination of the structural arrangement of atoms within Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0. 5. ZnGa 2 Se 4 crystallizes in the tetragonal space group I42m possessing a defective stannite structure 14 with a = 5.511 7͑3͒ Å and c = 10.964 3͑6͒ Å ͑see Fig. 1͒. The cations are surrounded by a tetrahedral environment of four Se atoms, with the anions located at the crystallographic positions 8i ͑x , x , z͒ with x ϳ 1/4 and z ϳ 1 / 8. Half of the Ga cations occupy the 2a ͑0, 0, 0͒ site, while the rest share the 4d position ͓ ͑0,1/2,1/4͒, ͑0,1/2,3/4͒ ͔ with the Zn atoms. Thus, the crystal structure exhibits a random disorder comprising all the Zn and half of the Ga atoms. The structure is defective in the sense that there are some vacancies which are ordered and located at the 2b ͑0,0,1/2͒ sites.
The other end member of the Zn 1−x Mn x Ga 2 Se 4 series is MnGa 2 Se 4 . The crystal structure of this magnetic material is rather similar to that reported for the Zn analog ͑see Fig. 1͒. The structural arrangement of the atoms corresponds to a defective chalcopyrite structure, tetragonal space group I4, with a = 5.677͑1͒ Å and c = 10.761͑2͒ Å. 13 The key difference between the crystal structure of the two end members of the series is the absence of cation disorder in MnGa 2 Se 4 . Thus, the Mn atoms are located in 2d ͑0,1/2,3/4͒ crystallographic positions and half of the Ga cations are in 2c sites ͑0,1/2,1/4͒. This forces the Se anions to move into a general position 8g, such that the x and y coordinates are no longer equal. The remainder of the Ga cations and the vacancies occupy the 2a ͑0, 0, 0͒ and 2b ͑0,0,1/2͒ lattice sites, respectively.
The results of the refinements of the crystal structure of Zn 1−x Mn x Ga 2 Se 4 with 0 ഛ x Ͻ 0.5 obtained using the Rietveld method are shown in Table I ͑see also Figs. 2 and 3. The central issue is the arrangement of the magnetic cations, which could be randomly distributed ͑as for the Zn ions in ZnGa 2 Se 4 ͒ or ordered ͑as in MnGa 2 Se 4 ͒. The neutron powder diffraction data show that all the materials under study retain the symmetry of the Zn derivative. Thus all of them crystallize in the space group I42m. While the reflection conditions for space groups I4 and I42m are the same, the x and y coordinates of the Se atom can be used when choosing between these space groups, since they are different for I4, but equal for I42m. The refinements performed on the Zn 1−x Mn x Ga 2 Se 4 with 0 ഛ x Ͻ 0.5 series using I4 symmetry give identical values for the x and y coordinates of the selenium atom within the experimental error.
The defective stannite and chalcopyrite structural arrangements can be derived from the well-known cubic zincblende structure by considering two different cations rather than one, introducing vacancies into the lattice, and doubling the unit-cell parameter c. This lowering of symmetry is accompanied by a relaxation of the anions with respect to the ideal zinc-blende positions at ͑1/4,1/4,1/8͒, etc. Considering the unit-cell dimensions, the evolution of the tetragonal parameters a and c with x is dissimilar in Zn 1−x Mn x Ga 2 Se 4 with 0 ഛ x Ͻ 0.5. While a increases with the Mn content, c remains essentially constant ͑see Table I͒. Interestingly, the decrease of c / a with x suggests a tendency to an increase in cation ordering as the Mn content increases. 21 Moreover, the decrease of c / a with x, from the value of 2 expected for the cubic zinc-blende structure, also indicates an increase of the tetragonal distortion as the concentration of magnetic atoms is increased.
As starting models for the distribution of cations, the Mn atoms were allowed to occupy the lattice sites available to cations ͑2a and 4d͒ and the vacant cation sites ͑2b͒. The refinement of the crystal structure showed that the Mn ions exclusively occupy the 4d lattice sites. Therefore, the magnetic ions occupy the same lattice sites as the Zn atoms in the range 0 Ͻ x Ͻ 0.5 ͑see Table I͒ and this configuration in which the Mn ions are disordered is more stable than the ones in which the cations are partially or totally ordered ͑as in the case of MnGa 2 Se 4 ͒. This fact significantly influences the magnetic properties of the system ͑see Sec. III B͒. The Mn content of each sample has been obtained by doubling the occupancy factor of this cation in the corresponding 4d lattice site ͑see Table I͒. The main bond distances are shown in Table II. The distance Zn, Mn, Ga͑4d͒ -Se increases smoothly with x. This is to be expected, since in this type of material the Mn-Se bond distance is larger than the Ga-Se and ͑Zn,Ga͒-Se distances. The bond distances Mn-Se and ͑Zn,Ga͒-Se have been reported to be 2.55͑2͒ and 2.4365͑5͒ Å, respectively, in MnGa 2 Se 4 ͑Ref. 13͒ and ZnGa 2 Se 4 . 14 The partial substitution of Zn by Mn does not change the population of the 2a lattice site and, as expected, the bond distance Ga͑2a͒-Se in Zn 1−x Mn x Ga 2 Se 4 with 0 ഛ x Ͻ 0.5 is not significantly affected by the substitution process. Its value is in agreement with that reported for ZnGa 2 Se 4 ͓2.4204͑10͒ Å͔ and MnGa 2 Se 4 ͓2.42͑2͒ Å͔.

B. Magnetic behavior
The dominant mechanism responsible for the magnetic interaction between Mn ions in DMSs has been found to be superexchange. 22,23 The evolution of the inverse of the dc magnetic susceptibility data, 1 / X, of Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5 as a function of temperature is presented in Fig.  4. These experiments indicate that the systems behave as paramagnets in the temperature range considered. The data have been fitted to a Curie-Weiss law X = C / ͑T − ͒ over the temperature range 130-250 K to obtain the Curie constant C and the temperature correction The latter takes a positive TABLE I. Mn content ͑x diff ͒, atomic coordinates for Se ͑Y = X͒ ͑Se at special position 8i; Mn, Zn, and Ga͑1͒ at special position 4d; Ga͑2͒ at special position 2a͒, occupancy factor of Mn ͑occupancy factor of Mn + occupancy factor of Zn+ occupancy factor of Ga͑1͒ = 1.0; Occ͑Ga͑1͒͒ = 0.5; Occ͑Zn͒ = 0.5− Occ͑Mn͒; Occ͑Ga͑2͒͒ = 1.0͒, lattice parameters, isotropic temperature factors, and conventional goodness-of-fit factors from Rietveld refinement of neutron-diffraction data obtained at 300 K for Zn 1−x Mn x Ga 2 Se 4 . Space group: tetragonal I42m.
Occ͑Mn͒ ITF͑Se͒  value in the case of predominantly ferromagnetic interactions and a negative value in the case of predominantly antiferromagnetic ones. Since C depends on the quantity of magnetic ions, 24 and therefore on x, it is possible to calculate the content of Mn of each sample from the measured magnetic data. The estimated accuracy of this method is ±5%. 25 Thus C = xNg 2 B 2 S͑S +1͒ /3k B , where x represents, in our case, the fraction of magnetic ions, N is Avogadro's constant, g is the Landé constant, B is the Bohr magneton, S is the spin of the magnetic ion ͑5 / 2 for Mn 2+ ͒, and k B is the Boltzmann constant. Everything is known in the above expression, except the g factor, which has been taken to be that of the Mn ion in MnGa 2 Se 4 . This g value can be easily calculated from the experimental Curie constant C for x =1.
The expression to calculate the Mn content of the Zn 1−x Mn x Ga 2 Se 4 series with 0 Ͻ x Ͻ 0.5 is the following: x = ͓␣C͑x͔͒ / ͓−␤C͑x͒ + C͑x =1͔͒, where C͑x͒ is the Curie constant of the dilution x in emu g −1 K −1 , C͑x =1͒ is the Curie constant of MnGa 2 Se 4 in emu mol −1 K −1 , and ␣ and ␤ determine the molecular weight of the series, M w ͑g mol −1 ͒, through the relationship M w = ␣ + ␤x. The above mathe-matical expression for x is of general application for a DMS. In the case under study, ␣ = 520.676 g mol −1 ͑the molecular weight corresponding to ZnGa 2 Se 4 ͒ and ␤ = −10.452 g mol −1 . The Weiss temperatures corresponding to the Zn 1−x Mn x Ga 2 Se 4 series have also been obtained from a Curie-Weiss law fit. Both x and values are shown in Table II.
Very different and C values can be found for a given series of diluted magnetic semiconductors, depending on the temperature range used for the fitting of the experimental susceptibility data. The knowledge of the x values for Zn 1−x Mn x Ga 2 Se 4 from neutron powder diffraction data can help in selecting the appropriate temperature range. These Mn contents can be used to calculate a Curie constant, which we have named C diff and the inverse of which gives the slope of the 1 / X vs T straight line. The slope associated to C diff should be equal, within the experimental error, to that determined by fitting the experimental magnetic data to 1 / X = − / C + T / C. We have used this agreement as a criterion to support the correctness of the temperature range used for the fitting and, therefore, the reliability of the obtained values. The agreement between the slopes is equivalent to that between the Mn contents calculated from neutron powder diffraction and magnetic data and is shown in Table II for The negative values of indicate predominantly antiferromagnetic interactions within Zn 1−x Mn x Ga 2 Se 4 . MnGa 2 Se 4 has been reported to order as an antiferromagnet below T = 6.4± 0.1 K with =−23±2 K. 15 The mathematical expression for the Weiss temperature is = ͓2S͑S +1͒ /3͔ ͚͑z i J i / k B ͒, where S is the spin of the magnetic ion ͑S =5/2 for Mn 2+ ͒, z represents the number of magnetic neighbors of a given reference ion, and J is the corresponding magnetic superexchange constant. 24 The index of the summation refers to the magnetic superexchange pathways with J 0. If z = xz 0 this expression can be simplified to ͑x͒ = x 0 , where 0 and z 0 refer to the undiluted compound. However, in the case of Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5, the probability of finding one magnetic ion neighboring another is given by instead of x, where is defined as the occupancy factor of the Mn ion on the 4d lattice site. Therefore, the Weiss temperature can be written as ͑͒ = ͓2S͑S +1͒ /3͔ ͚͑z 0,i J i / k B ͒ for that series, where z 0,i refers to the corresponding magnetically undiluted system and the values are taken from Table I ͓ = Occ͑Mn͔͒. Using this mathematical expression, a global superexchange magnetic constant ͚͑z 0,i J i / k B ͒ =−36±5 K can be estimated by fitting the experimental data for vs to a straight line. This global magnetic constant is about one order of magnitude higher than that reported for MnGa 2 Se 4 ͓͚͑z 0,i J i / k B ͒ =−3.9±2 K͔ 15 for which the magnetic superexchange pathways are of the type Mn-Se¯Se-Mn with antiferromagnetic constants of around J / k B = 0.1-0.5 K. However Mn-Se-Mn contacts must be also considered in the case of Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5. This is a magnetic pathway reported for zinc-blende structured Zn 1−x Mn x Se, which exhibits a related crystal structure, 22 with a corresponding superexchange constant of −12.2 K. 26 Therefore, the existence of higher magnetic interactions associated with shorter TABLE II. Main bond distances obtained from Rietveld refinement of neutron-diffraction data taken at 300 K for Zn 1−x Mn x Ga 2 Se 4 . Space group: tetragonal I42m. The Curie-Weiss constant, , and the manganese content, x mag , have been calculated from magnetic susceptibility data.  magnetic pathways is shown to play a significant role in the Zn 1−x Mn x Ga 2 Se 4 series when compared to MnGa 2 Se 4 . In the series under study, the magnetic constant between first nearest neighbors can be estimated using the global magnetic constant calculated above. Because only the order of magnitude of the superexchange constants for MnGa 2 Se 4 is known, 15 we have used the data published for zinc-blende Zn 1−x Mn x Se. 26 The exchange pathways are expected to be equivalent due to the similarity of the crystal structures. Those data for Zn 1−x Mn x Se provide J 1 / k B , the exchange constant corresponding to Mn-Se-Mn interactions ͑Mn-Mn distance= a / ͱ 2 Å, where a is the cubic unit-cell parameter͒, and three superexchange constants between next-nearest neighbors involving Mn-Se-Se-Mn-type contacts: J 2 / k B ͑Mn-Mn distance= a Å͒, J 3 / k B ͑Mn-Mn distance = a ͱ 3/2 Å͒, and J 4 / k B ͑Mn-Mn distance= a ͱ 2 Å͒. In order to simplify the model, we have also assumed that c /2Ϸ a and that J i / k B ͑i =1,2,3,4͒ does not change significantly with x. This hypothesis is supported by the evolution of the Mn-Mn distances as a function of x in Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5 ͑see Fig. 5͒. With these approximations, the Curie-Weiss temperature of this series, ͑͒ = ͓2S͑S +1͒ /3͔ ͚͑z 0,i J i / k B ͒, can be written as ͑͒ = ͓2S͑S +1͒ /3͔ ͓4J 1 / k B +6J 2 / k B +8J 3 / k B +8 J 4 / k B ͔. Using ͚͑z 0,i J i / k B ͒ = −36 K ͑from the previous paragraph͒ and the exchange constants reported for Zn 1−x Mn x Se ͑Ref 26͒ ͑J 2 / k B = −0.2 K, J 3 / k B = −0.1 K, and J 4 / k B = −0.4 K͒, a value of J 1 / k B = −7.7 K can be deduced from the expression ͚͑z 0,i J i / k B ͒ = ͓4J 1 / k B +6J 2 / k B +8J 3 / k B +8J 4 / k B ͔. In the case of the zinc-blende structure, J 1 / k B = −12.2 K. 26 Due to the approximations introduced, J 1 / k B = −7.7 K should, of course, be regarded as an estimate of the order of magnitude of J 1 / k B in Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5.
One interesting aspect of the series under study is that the lattice sites available to the magnetic cations are not the same in MnGa 2 Se 4 ͑2d͒ as in Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5 ͑4d͒. As a consequence, the magnetically undiluted system corresponding to Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5 is not MnGa 2 Se 4 but, rather, a hypothetical compound for which the magnetic atoms are distributed at random over the 4d lattice sites of space group I42m with 50% occupancy. The Curie-Weiss temperature of such a magnetically undiluted archetype system can be estimated using the expression = ͓2S͑S +1͒ /3͔ ͚͑z 0,i J i / k B ͒ with = 0.5 ͑that is, x =1͒ and the previously derived value ͚͑z 0,i J i / k B ͒ = −36 K. Thus, a value of ͑x =1,I42m͒ = −105 K can be calculated. As expected, the Curie-Weiss temperature of that hypothetical system, ͑x =1;I42m͒, is higher than that of the real undiluted compound MnGa 2 Se 4 , ͑MnGa 2 Se 4 ͒ =−23 K, 15 which exhibits a completely ordered distribution of magnetic ions. The reason is, once again, the creation of the superexchange pathway Mn-Se-Mn with a stronger associated magnetic constant in the case of I42m symmetry. A search of the literature to find a real magnetically undiluted compound in which the Mn ions are distributed at random over the 4d lattice sites of I42m symmetry ͑50% occupancy͒ shows that MnIn 2 Te 4 fulfils these requirements. Interestingly, the Curie-Weiss constant of this material has been found to be ͑MnIn 2 Te 4 ͒ =−88±10 K, 27 which compares well with that of our hypothetical system ͑x =1;I42m͒ = −105 K.

IV. CONCLUSIONS
Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5 retains the crystal symmetry of the parent ZnGa 2 Se 4 , with a defective stannite structure described by the tetragonal space group I42m. Although the magnetic cations are completely ordered onto a particular lattice site in MnGa 2 Se 4 , the controlled substitution of Zn by Mn in ZnGa 2 Se 4 ͑0 Ͻ x Ͻ 0.5͒ results in complete disorder of the magnetic ions over the corresponding crystallographic position. A tendency to cation ordering as the content of Mn increases is suggested by the decrease of the unit-cell parameter c / a with x.
The disordered distribution of the Mn ions has a profound effect on the magnetic behavior of Zn 1−x Mn x Ga 2 Se 4 with 0 Ͻ x Ͻ 0.5. The magnetically undiluted system corresponding to that series is not MnGa 2 Se 4 but, instead, a hypothetical compound for which the magnetic atoms are distributed at random over the 4d lattice sites of space group I42m with 50% occupancy. Moreover, the global magnetic interaction between manganese ions is found to be significantly higher than that expected for the classical magnetic dilution of the end member of the series MnGa 2 Se 4 .
The distance between the Mn ions along the various magnetic superexchange pathways is not significantly affected by the degree of magnetic dilution. This finding suggests that the magnetic superexchange constants do not vary significantly with x along the Zn 1−x Mn x Ga 2 Se 4 series with 0 Ͻ x Ͻ 0.5. In addition, a method of general application is provided for the selection of the appropriate range of temperature to calculate a correct Weiss constant in DMSs. The method is based on the precise knowledge of the concentration of magnetic ions within the sample and their influence on the Curie constant C.