The Zn_1−xPb_xCr₂Se₄—Single Crystals Obtained by Chemical Vapour Transport—Structure and Magnetic, Electrical, and Thermal Properties

Abstract

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 H_dc = 50 kOe.

1. Introduction

Magnetic and semiconductor spinels are of great interest because of their exotic phenomena and the fascinating ground states observed at the turn of the 20th and 21st centuries, such as [1]: behaviour of heavy fermions [2,3], complex spin order and spin dimerization [4,5,6], spin-orbital fluid [7,8], orbital glass [9], the coexistence of ferromagnetism and ferroelectricity [10,11,12], the so-called Jahn–Teller effect [13,14,15], and phenomena attributed to the competing exchange of ferromagnetism (FM) and anti-ferromagnetism (AFM) of almost equal strength, causing a strong bond frustration [16].

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−xCu_xCr₂Se₄ spinel series was particularly intensively studied, the edge spinels of which ZnCr₂Se₄ and CuCr₂Se₄ 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] ZnCr₂Se₄ 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 ZnCr₂Se_4, the spin order occurring at the Néel temperature T_N = 21 K [1] is an incommensurate spiral due to the narrow distances between Cr ions. Still, the larger ones cause ferromagnetic (FM) interactions confirmed by the positive Curie–Weiss temperature of θ = 90 K [21,22]. For this reason, differences in magnetic interactions between the nearest and farther neighbours in ZnCr₂Se₄ are related to the frustration of chemical bonds that play an essential role. The magnetic order is accompanied by structural transformation from cubic $Fd\overline{3}m$ to tetragonal I4₁/amd symmetry with a slight contraction along the c axis [24,25], manifested by sharp first-order anomalies at T_N on the specific heat curve [1].

Our group’s work has been done based on the ZnCr₂Se₄ matrix. This matrix has been doped with both non-magnetic ions, such as, for example, Al [26,27,28], Cu [17,18,19,29,30], Ga [17,18,31,32,33], In [18,33,34], Sn [35,36], and Ta [37], and with magnetic ones, such as, for example, Dy [38,39], Gd [40], Ho [41], Mn [42,43], Nd [44], and Ni [45]. All the above-mentioned doped spinels have two features in common. They are semiconductors and antiferromagnets, just like a matrix. An interesting feature is the appearance of the re-entrant-type spin-glass when non-magnetic Al [27] and Sn [36] ions are located in the octahedral sites of the spinel structure. It was also observed that Sn ions occupying octahedral sites reduce the n-type electrical conductivity by magnitude compared to samples in which they occupy tetrahedral ones [35]. This interesting phenomenon has been explained based on the quantum band model, which predicts a lowering of the Fermi level in the lowest Cr³⁺ Mott–Hubbard sub-band with a narrowed 3d³ t_2g band, which gives an increase in the energy gap when Sn³⁺ ions (spin defects) are replaced at octahedral sites instead of Cr³⁺ ones [35]. Interestingly, geometric spin-glass was observed when substituting Dy ions, a paramagnetic rare-earth ion in both tetrahedral and octahedral positions [39].

This work continues our research that involved introducing the p-block element into the crystal lattice ZnCr₂Se₄. Pb belongs to group 14 of the periodic table. Sn also belongs to this group. ZnCr₂Se₄ single crystals doped with tin showed the phenomenon of spin-glass [36]. It was assumed that the presence of Pb ions in the ZnCr₂Se₄ crystal lattice would affect the physicochemical properties of the parent compound. A single crystal growth model was developed using a computer programme. The structural, magnetic, electrical, and thermal properties were determined using various methods (XRD—X-ray Diffraction, SEM—Scanning Electron Microscopy, DSC—Differential Scanning Calorimetry, TG—Thermogravimetry, SQUID—Superconducting Quantum Interference Device, QD-PPMS—Quantum Design Physical Property Measurement System). This work presents the structural, magnetic, electrical, thermal, and specific heat properties of five monocrystalline samples doped with diamagnetic Pb²⁺ ions with a large ion radius instead of Zn²⁺ ones. The novelty of this work is the study of the ac magnetic susceptibility measured in an external dc magnetic field up to 50 kOe. According to our knowledge, such crystals have not been described in the literature.

2. Materials and Methods

High-purity elements, Zn, Pb, Cr, Se (5N, Sigma Aldrich, Sofia, Bulgaria), and anhydrous CrCl₃ (Sigma Aldrich), were applied to synthesise ZnCr₂Se₄ single crystals doped with Pb.

2.1. 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:

$$\Delta G_T^0=\Delta H_{298K}^0-T\Delta S_{298K}^0$$

(1)

where $\Delta H_{298K }^0$ and $\Delta S_{298K}^0$ 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 $\Delta G^0$ instead of the equilibrium constant K_p because these quantities are related to the formula:

$$\Delta G^0=-2.303RTlog{K}_p=\Delta H^0-T\Delta S^0,$$

(2)

where: R—gas constant; T—absolute temperature; K_p—equilibrium constant; $\Delta G^0$, $\Delta H^0$, $\Delta S^0$—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 $\Delta G^0$, 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 Δp_i, with the slightest possible temperature difference.

In the process of selenium chromite single crystal growth, anhydrous CrCl₃ is used, which dissociates at a temperature above 773 K according to the following reactions [46,47]:

2CrCl_3(g) = 2CrCl_2(g) + Cl_2(g),

CrCl_2(g) + Cl_2(g) = CrCl_4(g).

In a closed system, in a gas solution, apart from CrCl₃ and Cl₂, there is also CrCl₄. Therefore, the dissociation products should be taken into account when creating a set of all independent transport reactions that occur simultaneously in the ZnSe–PbSe–CrCl₃ system:

• 2ZnSe + 4CrCl_3(g) = 2ZnCl_2(g) + 4CrCl_2(g) + Se_2(g)
• 2ZnSe + 4CrCl_3(g) = Zn₂Cl_4(g) + 4CrCl_2(g) + Se_2(g)
• 2PbSe + 2CrCl_3(g) = 2PbCl(g) + 2CrCl_2(g) + Se_2(g)
• 2PbSe + 4CrCl_3(g) = 2PbCl_2(g) + 4CrCl_2(g) + Se_2(g)
• 2ZnSe + 2CrCl_4(g) = 2ZnCl_2(g) + 2CrCl_2(g) + Se_2(g)
• 2ZnSe + 2CrCl_4(g) = Zn₂Cl_4(g) + 2CrCl_2(g) + Se_2(g)
• 2PbSe + CrCl_4(g) = 2PbCl_(g) + CrCl_2(g) + Se_2(g)
• 2PbSe + 2CrCl_4(g) = 2PbCl_2(g) + 2CrCl_2(g) + Se_2(g)
• 2ZnSe + 2Cl_2(g) = 2ZnCl_2(g) + Se_2(g)
• 2ZnSe + 2Cl_2(g) = Zn₂Cl_4(g) + Se_2(g)
• 2PbSe + Cl_2(g) = 2PbCl_(g) + Se_2(g)
• 2PbSe + 2Cl_2(g) = 2PbCl_2(g) + Se_2(g)

These hypothetical reactions were used to calculate the temperature dependence of the thermodynamic parameters (ΔH⁰, ΔG⁰, ΔS⁰, and logK_p). The calculations were performed using the H.S.C. Chemistry 6 computer programme. Based on these calculations, the reaction conditions (dissolution zone temperatures and crystallisation and their difference (

Table 1. Conditions of growth and chemical composition of ZnCr₂Se₄:Pb single crystals.

No	Amounts of Substrates (mmol)		Td (K)	Tc (K)	ΔT (K)	% Weight				Chemical Formula
	nZnSe	nPbSe				Zn	Pb	Cr	Se
(1)	14.4	1.6	1223	1163	60	12.31 ± 0.09	3.23 ± 0.06	20.66 ± 0.01	63.71 ± 0.04	Zn0.94Pb0.06Cr2Se4
(2)	14.4	1.6	1133	1083	50	12.25 ± 0.08	2.95 ± 0.07	20.45 ± 0.02	64.54 ± 0.05	Zn0.92Pb0.07Cr2Se4
(3)	12.8	3.2	1203	1153	50	12.05 ± 0.06	3.43 ± 0.06	20.49 ± 0.05	64.10 ± 0.03	Zn0.92Pb0.09Cr2Se4
(4)	12.8	3.2	1203	1143	60	11.63 ± 0.02	4.51 ± 0.04	20.30 ± 0.06	63.56 ± 0.03	Zn0.89Pb0.11Cr2Se4
(5)	11.2	4.8	1203	1153	50	11.42 ± 0.03	4.98 ± 0.05	20.15 ± 0.03	63.42 ± 0.03	Zn0.88Pb0.12Cr2Se4

Chemical compositions were determined using Scanning Electron Microscopy. T_d is the temperature of the dissolution zone, T_c is the temperature of the crystallisation zone, and ΔT is the difference in temperatures between the dissolution and the crystallisation zones.

)) make it possible to obtain good-quality crystals.

2.2. 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).

2.3. 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.

2.4. 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]: ${\mu }_{eff}=2.828\sqrt{C}$, where C is the Curie constant. The magnetic super-exchange integrals for the first two coordination spheres, J₁ and J_2, were calculated using the Holland and Brown equations [50]: $J_1=\left(-9T_{\mathrm{N}}+\mathsf{\theta }\right)/60$ and $J_2=\left(3T_{\mathrm{N}}+\mathsf{\theta }\right)/120$. 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 $\mathsf{\sigma }={\mathsf{\sigma }}_0\exp (-\frac{{\mathrm{E}}_{\mathrm{a}}}{\mathrm{k}T})$, where k is the Boltzmann constant and σ₀ 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].

3. Results and Discussion

3.1. Growth of Single Crystals and Chemical Composition

Thermodynamic calculations indicate that, for the ZnSe–PbSe–CrCl₃ system, the simultaneous transport of ZnSe and PbSe will be realised using gaseous CrCl₃ and CrCl₄ (logK_p and ΔG⁰ values are close to zero in the selected temperature range: 1000–1400 K). It is shown in Figure 1 and Figure 2 that, for chemical reactions with CrCl₃ and CrCl_4, the enthalpy values are positive (ΔH⁰ > 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 (Cl₂) are negative (ΔH⁰ < 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–CrCl₃ system, ZnSe and PbSe are mainly transported by gaseous CrCl₃ and CrCl₄. 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–xPb_xCr₂Se₄ was carried out according to the reaction:

4(1 − x)ZnSe + 4xPbSe + 2CrCl₃ = Zn_1−xPb_xCr_xSe₄ + 3(1 − x)ZnCl₂ + 3xPbCl₂

for x = 0.1, 0.2, 0.3.

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 ZnCr₂Se₄ single crystals, we can state that a minor temperature difference (ΔT) is more favourable for crystal growth. Stoichiometric mixtures of the ZnSe, PbSe, and CrCl₃ 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.

3.2. Structural Study

Based on the structure of ZnCr₂Se₄, in which the Zn²⁺ ions occupy the tetrahedral position 8a: 1/8, 1/8, 1/8 (A-site), and the Cr³⁺ 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 Pb²⁺ ions in A-sites was assumed with coupled site occupancy factors (SOF) and constrained atomic displacements. The second model assumed that the Pb²⁺ ions would substitute the Cr³⁺ 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

Table 2. Structural parameters of the Zn_1-xPb_xCr₂Se₄ single crystals.

Chemical Formula	Lattice Parameter (Å)	Volume (Å3)	Density Calc. (Mg/m3)	Absorption Coeff. (mm−1)	Goodness of Fit on F2	R Parameters
						R1	wR2
Zn0.94Pb0.06Cr2Se4	10.4910(3)	1154.65(2)	5.682	32.837	1.150	0.0223	0.0662
Zn0.92Pb0.07Cr2Se4	10.4933(2)	1155.41(3)	5.689	32.883	1.205	0.0333	0.1114
Zn0.92Pb0.09Cr2Se4	10.4946(4)	1155.84(1)	5.709	32.856	1.175	0.0261	0.0839
Zn0.89Pb0.11Cr2Se4	10.4954(2)	1156.10(2)	5.757	33.841	1.196	0.0333	0.1114
Zn0.88Pb0.12Cr2Se4	10.4959(1)	1156.27(3)	5.772	33.852	1.191	0.0311	0.1012

and

Table 3. Atomic coordinates and equivalent isotropic displacement parameters of the Zn_1−xPb_xCr₂Se₄ single crystals. The Wyckoff positions of the atoms in the spinel structure are: Zn in 8 b (3/8, 3/8, 3/8); Cr/Ta in 16 c (½, ½, ½); and Se in 32 e (x, x, x). U (eq) is defined as one-third of the orthogonalized Uij tensor trace.

Spinel	Anion Parameter u	Site Occupation		Uiso (Å × 103)
		(A)	(B)	Zn/Pb	Cr	Se
(Zn0.94Pb0.06)[Cr2]Se4	0.2595(1)	0.94(1):0.06(1)	1.0	13(1)	7(1)	7(1)
(Zn0.92Pb0.07)[Cr2]Se4	0.2595(1)	0.92(2):0.07(1)	1.0	14(1)	6(1)	6(1)
(Zn0.92Pb0.09)[Cr2]Se4	0.2595(1)	0.92(1):0.09(1)	1.0	13(1)	7(1)	6(1)
(Zn0.89Pb0.11)[Cr2]Se4	0.2595(1)	0.89(2):0.11(2)	1.0	14(1)	5(1)	5(1)
(Zn0.88Pb0.12)[Cr2]Se4	0.2595(1)	0.87(1):0.13(2)	1.0	14(1)	5(1)	6(1)

. The observed slight increase in thermal shift at the A-site points out a static disturbance in the location of the Zn²⁺/Pb²⁺ pseudo-ion, which is usually caused by the difference in ionic charges and in ionic and covalent radii (r_i (Zn²⁺) = 0.60 Å, r_i (Cr³⁺) = 0.62 Å, r_i (Pb²⁺) = 0.98 Å and R_c (Zn²⁺) = 0.74 Å, R_c (Cr³⁺) = 0.76 Å, R_c (Pb²⁺) = 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−xPb_xCr₂Se₄. 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 ZnCr₂Se₄ (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 Zn²⁺ and Pb²⁺ (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. Selected interatomic distances (Å) and bond angles (deg) of the Zn_1-xPb_xCr₂Se₄ spinel single crystals.

Spinel	Bond Distances		Bond Angles
	Zn/Pb–Se	Cr–Se	Se–Zn/Pb–Se	Se–Cr–Se
(Zn0.94Pb0.06)[Cr2]Se4	2.4451(5)	2.5282(3)	109.5(0) × 6	180.0(0) × 3 94.61(13) × 6 85.39(13) × 6
(Zn0.92Pb0.07)[Cr2]Se4	2.4442(7)	2.5269(4)	109.5(0) × 6	180.0(0) × 3 94.61(2) × 6 85.39(2) × 6
(Zn0.92Pb0.09)[Cr2]Se4	2.4453(5)	2.5274(3)	109.5(0) × 6	180.0(0) × 3 94.62(14) × 6 85.38(14) × 6
(Zn0.89Pb0.11)[Cr2]Se4	2.4453(8)	2.5279(4)	109.5(0) × 6	180.0(0) × 3 94.62(2) × 6 82.38(2) × 6
(Zn0.88Pb0.12)[Cr2]Se4	2.4453(7)	2.5280(4)	109.5(0) × 6	180.0(0) × 3 94.61(2) × 6 85.39(2) × 6

), where the differences are insignificant [22]. The structure refinement parameters, the selected bond distances, and angles are shown in Table 3 and Table 4.

3.3. Electrical and Magnetic Properties

The results of the electrical measurements are shown in Figure 6 and

Table 5. Magnetic parameters of Zn_1−xPb_xCr₂Se₄ 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; J₁ and J₂ 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 ZnCr₂Se₄ matrix were taken from Refs. [1,21,22,31] for comparison.

Spinel	C	TN	θ	µeff	Ms(2K)	J1	J2	Hc1	Hc2	Ea
	(emu·K/mol)	(K)	(K)	(µB/f.u.)	(µB/f.u.)	(K)	(K)	(kOe)	(kOe)	(eV)
ZnCr2Se4	4.082	21	90	5.714	6.0	−1.65	1.28	10.0	65.0	0.135
Zn0.94Pb0.06Cr2Se4	5.477	22	74	6.618	5.325	−2.07	1.17	12.0	55.0	0.138
Zn0.92Pb0.07Cr2Se4	5.568	22	77	6.673	5.964	−1.95	1.23	12.0	56.0	0.161
Zn0.92Pb0.09Cr2Se4	4.727	22	81	6.149	5.960	−2.02	1.19	12.0	61.0	0.146
Zn0.89Pb0.11Cr2Se4	5.768	22	83	6.792	5.486	−1.92	1.24	12.0	55.0	0.177
Zn0.88Pb0.12Cr2Se4	5.898	22	84	6.868	5.251	−1.90	1.25	12.0	57.0	0.315

, and the magnetic ones are in Figure 7, Figure 8 and Figure 9 and Table 5 and

Table 6. Magnetic parameters of the Zn_1−xPb_xCr₂Se₄ 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 J₁ and J₂ are the super-exchange integrals for the first two coordination spheres.

Spinel	Hdc	C	TN	θ	Tm	µeff	J1	J2
	(kOe)	(emu·K/mol)	(K)	(K)	(K)	(µB/f.u.)	(K)	(K)
Zn0.94Pb0.06Cr2Se4	0	5.477	22	74	-	6.618	−2.07	1.17
	10	4.211	22	76	-	5.803	−1.73	1.13
	20	4.086	18	79	-	5.716	−1.38	1.11
	30	3.986	16	82	-	5.646	−1.03	1.08
	40	3.948	12	95	31.5	5.619	0.22	1.09
	50	3.863	6	108	46	5.558	0.90	1.05
Zn0.92Pb0.07Cr2Se4	0	5.568	22	77	-	6.673	−1.95	1.23
	10	4.182	20	78	-	5.783	−1.6	1.18
	20	4.102	19	81	-	5.728	−1.45	1.18
	30	4.003	16	84	-	5.658	−0.9	1.11
	40	3.988	12	96	33	5.648	−0.18	1.11
	50	3.976	7	99	44	5.639	0.60	1.00
Zn0.92Pb0.09Cr2Se4	0	4.727	22	81	-	6.149	−2.02	1.19
	10	3.922	21	82	-	5.601	−1.85	1.18
	20	3.711	20	84	-	5.448	−1.65	1.17
	30	3.681	16	85	-	5.426	−1.00	1.10
	40	3.602	12	97	33.5	5.367	−0.20	1.10
	50	3.639	6	108	46	5.369	0.90	1.05
Zn0.89Pb0.11Cr2Se4	0	5.768	22	83	-	6.792	−1.92	1.24
	10	4.293	20	84	-	5.859	−1.60	1.20
	20	4.182	19	85	-	5.783	−1.4	1.18
	30	4.101	16	86	-	5.727	−0.97	1.12
	40	4.056	12	91	34	5.695	−0.28	1.06
	50	4.002	6	97	48	5.657	0.72	0.96
Zn0.88Pb0.12Cr2Se4	0	5.898	22	84	-	6.868	−1.90	1.25
	10	3.794	20	86	-	5.508	−1.57	1.22
	20	3.781	19	88	-	5.499	−1.38	1.21
	30	3.762	16	90	-	5.485	−0.90	1.15
	40	3.703	12	94	36	5.442	−0.23	1.08
	50	3.689	6	98	48	5.432	0.73	0.97

. 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).

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 Pb²⁺ ions is much larger than that of Zn²⁺ 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].

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 ZnCr₂Se₄ 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 J₁ and J₂ 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 Cr³⁺ ions per molecule 3d³ 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 J₁ super-exchange integral for the first coordination sphere changes the sign from negative to positive, while the J₂ 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 ZnCr₂Se₄ matrix doped with Al [28], Ce, Ga, and In [33].

3.4. Specific Heat Studies

In Figure 10, the points represent the specific heat measured for samples with the formulae: Zn_0.92Pb_0.07Cr₂Se₄, Zn_0.92Pb_0.09Cr₂Se₄, Zn_0.89Pb_0.11Cr₂Se₄, and Zn_0.88Pb_0.12Cr₂Se₄.

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_0.92Pb_0.09Cr₂Se₄ exhibits quite a sharp peak in the vicinity of magnetic transition, whereas the remaining compositions show a broader two-peak-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_0.92Pb_0.07Cr₂Se₄ sample, which is a clear manifestation of antiferromagnetic order.

We also measured the resistivity for the Zn_0.88Pb_0.12Cr₂Se₄ 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 = ρ₀exp(Δ/kT) (ρ₀—resistivity at 273.15 K), where for an indirect bandgap, we obtained Δ ~0.88 eV. The inset shows the resistivity measured vs. magnetic field at room temperature.

The magnetoresistivity is positive and almost linear with the magnetic field. It is also relatively small, below 2% for 9 T.

3.5. Thermal Analysis

The thermal analysis examined two crystals of Zn_1−xPb_xCr₂Se₄ 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 ZnCr₂Se₄. For the sample Zn_0.94Pb_0.06Cr₂Se₄, two small endothermic peaks are observed at 711 °C and 733 °C. The observed peaks are in a similar position close to pure ZnCr₂Se₄ (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_0.88Pb_0.12Cr₂Se₄ 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 ZnCr₂Se₄.

4. Conclusions

In conclusion, we have demonstrated the family of Zn_1−xPb_xCr₂Se₄ (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 super-exchange 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 J₁-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.