The Zn1−xPbxCr2Se4—Single Crystals Obtained by Chemical Vapour Transport—Structure and Magnetic, Electrical, and Thermal Properties

Monocrystalline chalcogenide spinels ZnCr2Se4 are antiferromagnetic and semiconductor materials. They can be used to dope or alloy with related semiconducting spinels. Therefore, their Pb-doped display is expected to have unique properties and new potential applications. This paper presents the results of dc and ac magnetic measurements, including the critical fields visible on the magnetisation isotherms, electrical conductivity, and specific heat of the ZnCr2S4:Pb single crystals. These studies showed that substituting the diamagnetic Pb ion with a large ion radius for the Zn one leads to strong short-range ferromagnetic interactions in the entire temperature range and spin fluctuations in the paramagnetic region at Hdc = 50 kOe.

Many interesting properties were observed in the spinel series that fulfil Vegard's law in the full-concentration region. In the Katowice group at the end of the 20th century, the Zn 1−x Cu x Cr 2 Se 4 spinel series was particularly intensively studied, the edge spinels of which ZnCr 2 Se 4 and CuCr 2 Se 4 were respectively antiferromagnet and ferromagnet, and at the same time, a semiconductor and a conductor [17][18][19]. In the early 2000s, Parker et al. [20] discovered a transition from positive to negative magnetoresistance there.
Single crystals are used in modern, innovative technology and the electronic industry. The semiconducting [21] and antiferromagnetic [22] ZnCr 2 Se 4 spinel is a matrix for various diluted systems where the effects of the site disorder, lattice frustration, and random distribution of spin interactions [23] create novel potential applications in spin-based electronic technology. In ZnCr 2 Se 4, the spin order occurring at the Néel temperature

Computation of Thermodynamic Parameters for the Chemical Transport Reactions
The basis for the growth of single crystals by gaseous chemical transport is reversible transport reactions. The criterion determining the reaction's transport capacity is the order of magnitude of the equilibrium constant K p . The K p value should be close to 1 (logK p ≈ 0). The following dependence is used to assess the transport capacity of the reaction: where ∆H 0 298K and ∆S 0 298K are the corresponding changes of the reaction's standard and enthalpy and entropy. Free energy variations can be used to assess the transportability of the reaction ∆G 0 instead of the equilibrium constant K p because these quantities are related to the formula: where: R-gas constant; T-absolute temperature; K p -equilibrium constant; ∆G 0 , ∆H 0 , ∆S 0 -changes in the reaction's free energy, enthalpy, and entropy. Accompanying reactions related to the solvent or reaction products' dissociation may occur during the transport process. Therefore, all independent reactions occurring in the tested system should be considered. The selection of an appropriate transport substance involves finding such transport reactions for which the values of K p or ∆G 0 , in the selected temperature range, do not deviate too much from the equilibrium values. We are looking for a substance that will produce sizeable partial pressure differences ∆p i , with the slightest possible temperature difference.

Synthesis of ZnSe and PbSe
The polycrystalline ZnSe and PbSe were obtained by the high-temperature sintering of powders (1073 K) using a solid-state reaction in a high vacuum (10 −5 mbarr).

Crystal Structure and Chemical Composition
The X-ray diffraction measurement was carried out at 293(1) K. Five samples with different nominal Pb content were selected under a stereoscopic Zeiss optical microscope. A small crystal (diameter of about 0.1 mm) was mounted on a glass capillary. The data were collected using a SuperNova X-ray diffractometer with a microfocus X-ray tube, optimised multi-layer optics for Mo-Kα (λ = 0.71073 Å) radiation, and an Atlas CCD detector. Accurate cell parameters were determined and refined with CrysAlisPro software (Version 1.171.37.35, Agilent Technologies, Wrocław, Poland, 2014). Moreover, the CrysAlisPro program was used to integrate the collected data. The spinel structure (space group no. 227) was refined using the SHELXL-2013 program [48]. All atoms were refined with anisotropic displacement parameters. The chemical composition of the single crystals was studied using a Scanning Electron Microscope JEM 6480 equipped with an energy-dispersive X-ray spectrometer (SEM/EDS). The solid elements (Zn, Pb, Cr, Se) were used as standards to determine the mass % content of the elements in the sample. Measurements were carried out at 20 locations of the single crystal to determine the average chemical composition. Each of the measuring areas was approximately 50 × 30 µm. Then, the average chemical composition was calculated, which is shown in Table 1. The error bar represents the standard deviation. Relatively low values of the standard deviation indicate good homogeneity of the chemical composition.

Magnetic and Electrical Measurements
Dynamic ac magnetic susceptibility, χ ac , was measured in the temperature range 2-300 K and at an internal oscillating magnetic field H ac = 1 Oe, with an internal frequency f = 120 Hz taken at external static dc magnetic fields H dc = 0, 10, 20, 30, 40, and 50 kOe. Magnetisation isotherms, M(H), were measured at 2, 10, 20, 40, 60, and 300 K and in the static magnetic field up to 70 kOe. A Quantum Design MPMS-XL-7AC SQUID magnetometer (Quantum Design, San Diego, CA, USA) was used. The Néel (T N ) and Tm temperatures and the critical fields were defined as the extremes corresponding to the derivative of dχ ac /dT vs. T and dM/dH vs. H. The effective magnetic moment was calculated using the equation [37,42,49]: µ e f f = 2.828 √ C, where C is the Curie constant. The magnetic superexchange integrals for the first two coordination spheres, J 1 and J 2, were calculated using the Holland and Brown equations [50]: J 1 = (−9T N + θ)/60 and J 2 = (3T N + θ)/120. The electrical conductivity, σ(T), was measured by the dc method using a KEITHLEY 6517B Electrometer/High Resistance Meter (Keithley Instruments, L.L.C., Solon, OH, USA) in the temperature range of 77-400 K. The activation energy, E a , was determined below room temperature and in the temperature range of 300-400 K from the formula σ = σ 0 exp(− E a kT ), where k is the Boltzmann constant and σ 0 is the reference conductivity. The details of methods used to determine the electrical properties, thermal analysis, and specific heat are described in [37,39,41,44].

Growth of Single Crystals and Chemical Composition
Thermodynamic calculations indicate that, for the ZnSe-PbSe-CrCl 3 system, the simultaneous transport of ZnSe and PbSe will be realised using gaseous CrCl 3 and CrCl 4 (logK p and ∆G 0 values are close to zero in the selected temperature range: 1000-1400 K). It is shown in Figures 1 and 2 that, for chemical reactions with CrCl 3 and CrCl 4, the enthalpy values are positive (∆H 0 > 0), which indicates that the transport takes place from a higher temperature to a lower temperature in the direction of the formation of products.
On the other hand, the enthalpy values for the transport of reactions with chlorine (Cl 2 ) are negative (∆H 0 < 0), which indicates that the reaction equilibrium is shifted towards the reactants. Based on these calculations, it has been confirmed that in the ZnSe-PbSe-CrCl 3 system, ZnSe and PbSe are mainly transported by gaseous CrCl 3 and CrCl 4 . Values of logK a are close to zero in the selected temperature range: 1000-1400 K ( Figure 3).   Synthesis of single crystals of Zn 1-x Pb x Cr 2 Se 4 was carried out according to the reaction: Thermodynamic calculations determined the reaction conditions, such as dissolution and crystallisation temperatures, and their difference. The crystallisation zone temperature was between 1083 and 1153 K. The dissolution zone was between 1133 and 1233 K. Their difference was 50-70 K. Based on our experience with the growth of doped ZnCr 2 Se 4 single crystals, we can state that a minor temperature difference (∆T) is more favourable for crystal growth. Stoichiometric mixtures of the ZnSe, PbSe, and CrCl 3 were introduced to quartz ampoules (length-200 mm, inner diameter-20 mm) evacuated to 10 −5 mbarr. The sealed ampoules were placed in a two-zone tubular furnace and heated for 450 h. After heating, the ampoules were cooled at about 70 degrees per hour. Such a preparation process for the growth of single crystals allowed for obtaining single crystals of good quality ( Figure 4). Five single crystals with various Pb content were chosen for X-ray diffraction measurements. The reaction conditions and results of SEM analysis, together with determined chemical formulae, are shown in Table 1.

Structural Study
Based on the structure of ZnCr2Se4, in which the Zn 2+ ions occupy the tetrahedral position 8a: 1/8, 1/8, 1/8 (A-site), and the Cr 3+ ions occupy the position 16d: ½ ½, ½ (B-site), according to the spinel structure, we considered two models of cation distribution. In the first model, the presence of Pb 2+ ions in A-sites was assumed with coupled site occupancy factors (SOF) and constrained atomic displacements. The second model assumed that the Pb 2+ ions would substitute the Cr 3+ ones. Because of the strong correlation, the SOFs for Cr and Pb were refined separately in alternative calculations. The as-

Structural Study
Based on the structure of ZnCr2Se4, in which the Zn 2+ ions occupy the tetrahedral position 8a: 1/8, 1/8, 1/8 (A-site), and the Cr 3+ ions occupy the position 16d: ½ ½, ½ (B-site), according to the spinel structure, we considered two models of cation distribution. In the first model, the presence of Pb 2+ ions in A-sites was assumed with coupled site occupancy factors (SOF) and constrained atomic displacements. The second model assumed that the Pb 2+ ions would substitute the Cr 3+ ones. Because of the strong correlation, the SOFs for Cr and Pb were refined separately in alternative calculations. The as-

Structural Study
Based on the structure of ZnCr2Se4, in which the Zn 2+ ions occupy the tetrahedral position 8a: 1/8, 1/8, 1/8 (A-site), and the Cr 3+ ions occupy the position 16d: ½ ½, ½ (B-site), according to the spinel structure, we considered two models of cation distribution. In the first model, the presence of Pb 2+ ions in A-sites was assumed with coupled site occupancy factors (SOF) and constrained atomic displacements. The second model assumed that the Pb 2+ ions would substitute the Cr 3+ ones. Because of the strong correlation, the SOFs for Cr and Pb were refined separately in alternative calculations. The as-(B-site), according to the spinel structure, we considered two models of cation distribution. In the first model, the presence of Pb 2+ ions in A-sites was assumed with coupled site occupancy factors (SOF) and constrained atomic displacements. The second model assumed that the Pb 2+ ions would substitute the Cr 3+ ones. Because of the strong correlation, the SOFs for Cr and Pb were refined separately in alternative calculations. The assumed oxidation state of Pb and the correlations of the refined parameters caused the rejection of the second model. Based on this refinement, the first model allowed us to obtain the acceptable atomic displacement parameters and SOFs. We could describe the general chemical formula for obtained single crystals as presented in Tables 2 and 3. The observed slight increase in thermal shift at the A-site points out a static disturbance in the location of the Zn 2+ /Pb 2+ pseudo-ion, which is usually caused by the difference in ionic charges and in ionic and covalent radii (r i (Zn 2+ ) = 0.60 Å, r i (Cr 3+ ) = 0.62 Å, r i (Pb 2+ ) = 0.98 Å and R c (Zn 2+ ) = 0.74 Å, R c (Cr 3+ ) = 0.76 Å, R c (Pb 2+ ) = 1.12 Å, respectively) [51].   The structural study confirmed the chemical composition obtained from SEM. The general formula of obtained single crystals can be described as Zn 1−x Pb x Cr 2 Se 4 . The Pb ions substituted the Zn ones. The single crystals crystallised in a cubic system with space group Fd-3m (No. 227, Z = 8). The lattice parameters of the obtained single crystals are slightly larger than pure ZnCr 2 Se 4 (a = 10.489 Å) and enhanced with the increase in lead content ( Table 2). The increase in lattice parameters is connected with the difference in ionic radii between Zn 2+ and Pb 2+ ( Figure 5). The positional parameter of Se (u) is a measure of the anion sublattice distortion from the cubic-close packing. The value of u is the same for all obtained single crystals. Its value increases slightly compared to the ideal value of u = 0.250 (Table 3). The admixture of lead does not significantly affect the u-value. The same is observed in the metal-metal and metal-selenium distances (Table 4), where the differences are insignificant [22]. The structure refinement parameters, the selected bond distances, and angles are shown in Tables 3 and 4.

Electrical and Magnetic Properties
The results of the electrical measurements are shown in Figure 6 and Table 5, and the magnetic ones are in Figures 7-9 and Tables 5 and 6. Electrical conductivity measurements ( Figure 6) showed semiconducting properties in the temperature range of 77-400 K for all single crystals under study. Two regions of Arrhenius type electrical conductivity are visible: the extrinsic one with the activation energy of E a~0 .08 eV below room temperature, independent of the Pb content in the sample, and the intrinsic one with E a increasing in the range of 0.138-0.315 eV above room temperature with increasing Pb content (Table 5).  Table 5. Magnetic parameters of Zn 1−x Pb x Cr 2 Se 4 single crystals recorded in an internal oscillating magnetic field H ac = 1 Oe with an internal frequency f = 120 Hz and with a zero external static magnetic field: C is the Curie constant; T N and θ are the Néel and Curie-Weiss temperatures, respectively; µ eff is the effective magnetic moment; J 1 and J 2 are the super-exchange integrals for the first two coordination spheres. H c1 and H c2 are the critical fields measured at the static magnetic field up to 70 kOe. E a is the energy activation measured in the temperature range of 300-400 K. Experimental data for the ZnCr 2 Se 4 matrix were taken from Refs. [1,21,22,31] Table 6. Magnetic parameters of the Zn 1−x Pb x Cr 2 Se 4 single crystals recorded in an internal oscillating magnetic field H ac = 1 Oe with an internal frequency f = 120 Hz and taken at external static magnetic fields H dc = 0, 10, 20, 30, 40, and 50 kOe: C is the Curie constant; T N , θ, and T m are the Néel, Curie-Weiss, and the spin fluctuation temperatures, respectively; µ eff is the effective magnetic moment; and J 1 and J 2 are the super-exchange integrals for the first two coordination spheres. The increase in activation energy with the increase in lead content may be caused by the appearance of cationic vacancies (acting as double acceptors) because the radius of Pb 2+ ions is much larger than that of Zn 2+ ones [51]. It may result in acceptor vacancy levels in the energy gap. The activation energies mentioned above correlate well with E a for the matrix [31] and with its admixtures, such as tin [35], tantalum [37], and dysprosium [22].

Spinel
The temperature dependencies of ac magnetic susceptibility χ ac , recorded in an internal oscillating magnetic field H ac = 1 Oe with an internal frequency f = 120 Hz and with a zero external static magnetic field (Figure 7a-e), show AFM order with the Néel temperature T N = 22 K and the positive Curie-Weiss temperature (θ) slightly changing from 74 K to 84 K as the Pb content increases ( Table 5).
The θ values are much lower than the literature data published in [21,22] and slightly lower than in [1], where θ is equal to 115 K and 90 K, respectively. Thus, only the T N temperature values are close to this temperature's values for the ZnCr 2 Se 4 matrix [1,21,22]. It means that the long-range AFM interactions are slightly more substantial. In contrast, the short-range FM ones are somewhat weaker in the sample containing Pb, which is confirmed by both the long bond lengths (Table 4) and the values of the J 1 and J 2 super-exchange integrals for the first two coordination spheres (Table 5). A significant difference was observed between the effective magnetic moment (µ eff ) and the effective number of Bohr magnetons, p eff = 5.477, i.e., the spin contribution to the magnetic moment for Cr 3+ ions per molecule 3d 3 configuration. It may suggest that the orbital magnetic contribution has not been quenched at low amplitude, H ac = 1 Oe, of the applied oscillating magnetic field.
The magnetisation measurements (Figure 8a-e) show that the magnetic saturation of the studied samples is close to the value of 6 µ B /f.u. for the matrix; the first critical field is kept constant at H c1 = 12 kOe, and the second critical field is around H c2 = 57 kOe ( Table 5). The hysteresis loops have zero-field coercivity and zero remanences (Figure 8f).
The temperature dependencies of ac magnetic susceptibility χ ac , recorded in an internal oscillating magnetic field H ac = 1 Oe with an internal frequency f = 120 Hz and taken at external static magnetic fields H dc = 0, 10, 20, 30, 40, and 50 kOe, are shown in Figure 9a-e. With the increase in H dc , we observed a shift of T N towards lower temperatures and of θ towards higher ones ( Table 6).
A strong magnetic field weakens the AFM order and strengthens the FM one. Moreover, the Curie constant, C, and the effective moment µ eff , are close to the values typical for a chromium ion per molecule. For H dc = 50 kOe, the J 1 super-exchange integral for the first coordination sphere changes the sign from negative to positive, while the J 2 integral remains positive (Table 6). It means that the short-range FM interaction extends over the entire temperature range. The χ ac (T) curves above T N in the paramagnetic region show characteristic wide maxima at T m = 32-36 and 44-48 K in the fields H dc = 40 and 50 kOe, respectively. These broad maxima of ac magnetic susceptibility may be due to the spin fluctuations that occur due to the amplification of short-range FM interactions by a static magnetic field, which is opposed by the thermal energy kT. Similar maxima were found in the ZnCr 2 Se 4 matrix doped with Al [28], Ce, Ga, and In [33].

Specific Heat Studies
In Figure 10  We see that the Debye model fits the experimental data well for all the samples, with the Debye temperatures obtained from the fits 163-177 K, with no apparent dependence of Debye temperature on the Pb concentration. The number of atoms obtained from the fit lies within 7.2-9.5 for the investigated samples, without clear dependence on the Pb concentration. It can be explained by the fact that the change in sample composition is minute, and the changes in Zn concentration accompany the changes in Pb concentration. The upper inset of Figure 10 shows the specific heat in a narrow region around the magnetic ordering temperature. The sample Zn 0.92 Pb 0.09 Cr 2 Se 4 exhibits quite a sharp peak in the vicinity of magnetic transition, whereas the remaining compositions show a broader twopeak-like structure. This is most likely caused by the disorder induced by the Pb substitution and the lack of full site occupancy of the Zn site. The bottom inset of Figure 10 shows the shift of the magnetic transition with the increase in magnetic field for the Zn 0.92 Pb 0.07 Cr 2 Se 4 sample, which is a clear manifestation of antiferromagnetic order.
We also measured the resistivity for the Zn 0.88 Pb 0.12 Cr 2 Se 4 sample (the sample with the highest Pb concentration) with the resistivity option in the PPMS instrument ( Figure 11). Below 120 K, the resistivity exceeded the range of the instrument. However, in the 120-300 K range, the resistivity can be well-fitted with an activation law ρ = ρ 0 exp(∆/kT) (ρ 0 -resistivity at 273.15 K), where for an indirect bandgap, we obtained ∆~0.88 eV. The inset shows the resistivity measured vs. magnetic field at room temperature. Figure 11. The resistivity of the Zn 0.88 Pb 0.12 Cr 2 Se 4 single crystal measured vs. temperature (central Figure) and vs. field at T = 300 K (inset). The Zn 0.88 Pb 0.12 Cr 2 Se 4 sample was tiny. Hence, we do not provide the absolute value of the resistivity.
The magnetoresistivity is positive and almost linear with the magnetic field. It is also relatively small, below 2% for 9 T.

Thermal Analysis
The thermal analysis examined two crystals of Zn 1−x Pb x Cr 2 Se 4 containing minimum (0.06) and maximum (0.12) lead amounts. The DSC and TG curves are depicted in Figure 12.
The shape of the DSC curves is similar to pure ZnCr 2 Se 4 . For the sample Zn 0.94 Pb 0.06 Cr 2 Se 4 , two small endothermic peaks are observed at 711 • C and 733 • C. The observed peaks are in a similar position close to pure ZnCr 2 Se 4 (755 • C) [44]. These peaks indicate that the melting process of the sample starts at about 711 • C. The peak confirms it on the DTG curve. However, the sample Zn 0.88 Pb 0.12 Cr 2 Se 4 lacks typical endothermic peaks. On the TG curves for both samples, a few percent increase in weight is visible in the first step. This phenomenon can indicate that the investigated sample absorbs an inert gas during the heating or solid-state-gas reaction. The mass loss is observed with increasing temperature on the DTG curve. It is an endothermic reaction, as indicated by the downward deflection of the DSC curve. The partial destruction of investigated samples takes place at about 1300 • C. It suggests that the samples doped with Pb are thermally more stable than pure ZnCr 2 Se 4 .

Conclusions
In conclusion, we have demonstrated the family of Zn 1−x Pb x Cr 2 Se 4 (x = 0.06, 0.07, 0.09, 0.11, 0.12) spinel single crystals showing semiconducting and AFM behaviour. These single crystals were successfully obtained using chemical vapour transport. The technology of crystal growth was based on thermodynamic calculations. XRD analysis showed that these crystals possess a cubic structure (SG: Fd3m). Thermal analysis showed that the single crystals under study are thermally stable up to 1300 • C.
With the increase in the external magnetic field, a shift of T N and the specific heat peak towards lower temperatures and θ towards higher ones was observed, as well as a strong weakening of long-range AFM interactions visible in the reduction of the superexchange integral for the first coordination sphere and the appearance of spin fluctuations in the paramagnetic region, visible in magnetic fields of 40 and 50 kOe. Below T N, the magnetic field dependence of magnetisation, M(H), showed two peculiarities at critical fields H c1 = 12 and H c2 = 57 kOe and a change of the sign of the J 1 -integral from negative to positive at H dc = 50 kOe, which causes the short-range FM interaction to extend over the entire temperature range. Finally, we can conclude that doped chalcogenide spinels are suitable materials for application in many technological areas, such as magnetic materials for capacitors, transformers, or components in electronic products.