The S = 1 spinel compound ZnV ₂ O ₄ has attracted much interest in recent years due to its novel applications in the energy area such as hydrogen storage, ^{[ 1 – 4 ]} as well as the unique physical properties of 3d electrons arising from strong coupling among charge, spin, orbit, and lattice. ^{[ 5 ]} At room temperature, the V ³⁺ ions form a corner-sharing, geometrically frustrated, pyrochlore network and are octahedrally coordinated by six oxygen ions, while the Zn ²⁺ ions are located in four tetrahedral oxygen ions, presenting the cubic symmetry Fd 3̅ M . ^{[ 6 ]} With temperature dropping, the symmetry of the lattice is broken by a structural transition induced by the Jahn–Teller effect. ^{[ 6 ]} Such transition leading by a lattice distortion was originally observed by Ueda’s team via peak splitting in the x-ray diffraction (XRD) pattern around the transition temperature T _S = 50 K. ^{[ 7 ]} They found that ZnV ₂ O ₄ is cubic above 50 K while tetragonal below 48 K with a compression along the c axis ( c / a ≈ 0.994). ^{[ 7 ]} Accompanied by this structural transition, a kind of orbital ordering emerges at the same temperature on account of the split of the t _2g level and the reoccupation of two lower energy levels by two electrons of V ³⁺ . ^{[ 8 ]} Tsunetsugu and Motome believed that the Coulomb and exchange interaction between V ³⁺ ’s played a decisive role, thus one electron of each V ³⁺ invariably occupied d _xy and another one occupied d _zx and d _yz alternatively. ^{[ 9 , 10 ]} However, based on the relativistic spin–orbit coupling effect, Tchernyshyov proposed a ferro-type orbital order with complex orbital state d _xz ± id _yz consistent with the system symmetry I 4 ₁ / amd . ^{[ 11 ]} On the other hand, neutron scattering experiments and the susceptibility result witnessed that a Néel transition from paramagnetism (PM) to antiferromagnetism (AFM) at transition temperature T _M = 40 K exists in ZnV ₂ O ₄ as well. ^{[ 7 , 12 , 13 ]} The configuration of such antiferromagnetic spin chains is still ambiguous owing to the geometric frustration and the spin–orbital coupling. Those two transitions were obviously observed in temperature-dependent susceptibility and specific heat by Vasiliev et al . ^{[ 14 ]} Although the two transition temperatures obtained by them, 45 K and 31 K, are lower than those identified in XRD and neutron scattering experiments, Vasiliev’s work ^{[ 14 ]} revealed that the structural and orbital transition is a first-order transition, while the AFM transition is a second-order one.

As is well known, the second-order transition is a continuous transition, whose critical behaviors and universal classifications could provide a deep insight into such transition, as well as the inherent relations between the transition and various interactions. ^{[ 15 ]} Thus, a lot of investigations of critical behaviors have been carried out for the second-order phase transitions with ferromagnetic and antiferromagnetic orders, as well as superconductive orders. However, few investigations into the critical behaviors of ferromagnetic or antiferromagnetic transition with the coexistence of spin–spin interaction and spin–orbital couplings were reported, especially in AB ₂ X ₄ spinel compounds. The critical behaviors of paramagnetic–ferrimagnetic and paramagnetic–ferromagnetic transitions with the competitions between spin, orbital, and lattice degrees of freedom were studied in MnV ₂ O ₄ and CdCr ₂ S ₄ by Zhang’s group and Lou’s group, respectively. ^{[ 16 , 17 ]} They both observed the discrepancy of critical exponents with respect to either the Heisenberg model or the mean field theory, and revealed some possible effects of spin–orbital and spin–phonon couplings on the critical phenomena. ^{[ 17 ]} Different from MnV ₂ O ₄ and CdCr ₂ S ₄ , the second-order phase transition in ZnV ₂ O ₄ is a paramagnetic–antiferromagnetic transition. Moreover, for MnV ₂ O ₄ , the temperature of structural transition accompanied by the orbital ordering ( T _S ) is lower than the magnetic transition temperature T _M , ^{[ 16 ]} while for ZnV ₂ O ₄ , T _S is higher than T _M . This indicates that the AFM transition in ZnV ₂ O ₄ is in the background of the orbital order. Obviously, it is significant to check the critical behavior of this kind of AFM transition accompanied with orbital order.

In this paper, the phase transitions and the critical behaviors of ZnV ₂ O ₄ are investigated via susceptibility and specific heat measurements. The critical exponent and the critical fluctuation region accompanying the second-order AFM transition are obtained. An inconsistency of universal classification identified by either the critical exponent or the critical amplitude ratio is found in this compound. Moreover, the derivative of pressure with respect to temperature around the first-order structural transition is derived from the Clausius–Clapayron equation.

The polycrystalline sample of ZnV ₂ O ₄ was synthesized by solid-phase reaction using high purity powders of ZnO (Sinopharm, 99.99%) and V ₂ O ₅ (Sinopharm, 99%). The starting powders were first mixed thoroughly at stoichiometric ratio; then the mixed powder was pressed into pellets and annealed at 800 °C for 24 h under H ₂ /N ₂ (concentration of H ₂ was 5%) mixed gas atmosphere. The obtained sample was a black bulk ceramic.

The crystal structure and phase purity of the sample were investigated by XRD by using Cu- Kα radiation (D8 Advanced, Bruker AXS, Germany) with 2 θ changing from 20° to 80° at room temperature. The temperature-dependent susceptibility and specific heat measurements were carried out on a physical properties measurement system (PPMS 6000, Quantum Design, USA). Specifically, the susceptibility against temperature ( χ – T ) was measured in a warming process from 2 K to 300 K under an applied field of 1 T after zero field cooling. The temperature-dependent specific heat ( C _P – T ) was measured between 2 K to 200 K in both cooling and warming processes without field. The data of specific heat were collected every 0.1 K in ranges of 2 K to 10 K and 30 K to 60 K, and every 0.01 K in the AFM transition region with a temperature range from 36 K to 39 K.

The structure of the ZnV ₂ O ₄ sample is characterized by XRD at room temperature. The XRD pattern is shown in Fig. 1 ; all characteristic peaks perfectly match the standard data without any redundant peak. ^{[ 18 ]} The obtained average lattice constant of our sample is 8.399 Å, which is very close to the standard value ( α = 8.409 Å) of this compound. ^{[ 18 ]}

Fig. 1.

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	Fig. 1. The experimental XRD pattern (upper) and the standard characteristic peaks of ZnV 2 O 4 (lower).

The temperature-dependent susceptibility of the sample is exhibited in Fig. 2(a) . Our χ – T curve shows similar characteristics to those obtained by Vasiliev. ^{[ 14 ]} Two obvious peaks locate at T _S = 50.1 K and T _M = 37.9 K, which correspond to the structural transition and the magnetic transition, respectively. The transition temperatures are close to those obtained by Ueda and Lee from the susceptibility measurement of ZnV ₂ O ₄ , T _S = 50 K and T _M = 40 K. ^{[ 7 , 19 ]}

Fig. 2.

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	Fig. 2. (a) Temperature-dependent susceptibility of ZnV 2 O 4 by ZFC-warming measurement. (b) Temperature-dependent specific heat of ZnV 2 O 4 by warming (solid red dots) and cooling (solid green squares) measurement. The inset shows C P / T vs. T for obviously displaying the peak and bump.

Figure 2(b) shows the temperature dependence of specific heat, where the solid red dots correspond to C _P measured in the cooling process, and the solid green squares represent those in the warming process. When T < 30 K, C _P monotonously decreases with the temperature dropping. When 30 K ≤ T ≤ 80 K, a sharp peak around 50 K and a small bump around 35 K could be observed, corresponding to the aforementioned structural and orbital transition and magnetic transition, respectively. The peak and the bump can be more easily viewed in the C _P / T vs. T curve as displayed in the inset of Fig. 2(b) . The C _P of ZnV ₂ O ₄ in this paper is of the same order of magnitude as that obtained by Vasiliev, ^{[ 14 ]} but the transition temperatures we observed are closer to those obtained by Ueda et al and Lee et al through the temperature-dependent susceptibility. ^{[ 7 , 19 ]} Around 50 K, there is an obvious hysteresis between C _P vs. T curves obtained in cooling and warming processes, indicating that the structural transition accompanied with orbital order is a first-order transition; while at 35 K, the C _P curves are in good coincidence, implying that the antiferromagnetic transition is a second-order one. It is worth mentioning here that the deviation between cooling and warming C _P curves around 200 K may be from the thermal disturbance during the measurement process.

In the following parts, C _P will be analyzed in detail in three different temperature regions: 1) the low temperature region with T < 30 K, 2) the first-order structural and orbital transition region around T _S = 50 K, and 3) the second-order AFM transition region around T _M = 35 K.

3.1. Low temperature region

At low temperature, only contributions from electrons, phonons, and magnons need to be considered. Therefore, C _P can be expressed as ^{[ 20 ]}

where γ , β , and δ are electrons, phonons, and magnons coefficients, and T is the temperature. Equation ( 1 ) can be rewritten as

According to the above equation, C _P / T should be linearly depending on T ² . Figure 3 shows C _P / T vs. T ² in the low temperature region. As can be seen, in temperature range of 2–5 K, C _P / T – T ² shows a good linear relationship.

Fig. 3.

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	Fig. 3. Linear relationship between C P / T and T 2 in temperature range of 2–5 K.

Through linear fitting, γ and ( β + δ ) are obtained. The fitting results are the same for both cooling and warming processes because they are identical in the considered temperature range. The related parameters obtained from fitting are γ = 5.85 × 10 ⁻³ J/mol·K ² and β′ = ( β + δ ) = 0.983 × 10 ⁻³ J/mol·K ⁴ . The γ of ZnV ₂ O ₄ is lower than that of MnV ₂ O ₄ (12 × 10 ⁻³ J/mol·K ² ) reported by Myung-Whun et al . ^{[ 21 ]} In MnV ₂ O ₄ , not only 3d electrons of V ³⁺ contribute to C _P , those of Mn ²⁺ may also have significant contributions. This might be the reason for MnV ₂ O ₄ to have larger γ compared with ZnV ₂ O ₄ . The γ can be related to Fermi energy ε _F and the density of states near the Fermi surface D ( ε _F ) of ZnV ₂ O ₄ as follows: ^{[ 22 ]}

where N _A and k _B are the Avogadro constant and the Boltzmann constant, respectively. Based on γ obtained by fitting, ε _F and D ( ε _F ) are derived to be 2.42 eV and 2.48 eV ⁻¹ , respectively. The relation between β and Debye Temperature Θ _D is given by ^{[ 22 ]}

Since β ≤ β′ = 0.983 × 10 ⁻³ J/mol·K ⁴ , the low limit of ZnV ₂ O ₄ ’s Θ _D is estimated to be 240 K, which is deviated from 330 K obtained by Vasiliev et al. The reason for this difference is that Vasiliev fitted the data in a larger temperature range. According to our results, as the temperature goes up, the C _P / T vs. T ² curve significantly deviates from a straight line.

3.2. Structural and orbital transition region

The peaks around 50 K in C _P curves for both cooling and warming processes correspond to the structural transition. Excess specific heat Δ C _P is obtained by subtracting the regular contribution from C _P and shown in Fig. 4(a) . The hysteresis of Δ C _P between cooling and warming processes is more obvious here, which is strong evidence indicating that the transition here is a first-order one. The transition temperatures T _S for warming and cooling processes are 50.1 K and 50.3 K, respectively. The T _S for the warming process is exactly the same as that obtained from the temperature dependent susceptibility (shown in Fig. 2 ). Through the Δ C _P curve, the enthalpy change Δ H and entropy change Δ S could be calculated and the results are shown in Table 1 . The Clausius–Clapayron equation on this transition could be written as ^{[ 23 ]}

where P is the pressure, Δ H is the enthalpy change equal to the latent heat, T _S is the transition temperature, and v ₁ and v ₂ are the specific volumes before and after the transition, respectively. The v ₂ − v ₁ could be derived from the lattice constant and the compression ratio. ^{[ 7 ]} At T _S , the derivative of pressure with respect to temperature for temperature decreasing, d P /d T = 9.88 × 10 ⁶ Pa/K, is greater than that for temperature increasing (8.58 × 10 ⁶ Pa/K).

Fig. 4.

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	Fig. 4. (a) The excess specific heat of ZnV 2 O 4 at structural transition for warming (solid red dots) and cooling (solid green squares) processes. (b) The excess specific heat of CdV 2 O 4 at AFM transition for both processes.

Table 1.

Table 1.

Table 1. Enthalpy change, entropy change, and specific volume difference after and before structural transition of ZnV 2 O 4 in warming and cooling measurement. . Δ H /J·mol −1 Δ S /J·mol −1 ·K −1 Δ V /m 3 ·mol −1 Warming 115.68 2.24 2.69 × 10 −7 Cooling −133.63 −2.55 −2.69 × 10 −7

Table 1. Enthalpy change, entropy change, and specific volume difference after and before structural transition of ZnV 2 O 4 in warming and cooling measurement. .

3.3. Antiferromagnetic transition region

The bump in the C _P curve shown in Fig. 2(b) in the temperature range of 30 K to 40 K corresponds to the AFM transition of ZnV ₂ O ₄ . The coincidence of the C _P – T curves for both cooling and warming processes in this temperature range confirms that this transition is a second-order phase transition. Since the orbits have already been ordered, besides the spin–spin interaction, the spin–orbital interaction also exists, and may have a significant effect on the critical behaviors of this transition. The excess specific heat of this transition is obtained by using the same method described previously. The Δ C _P vs. T curve for the AFM transition region is shown in Fig. 4(b) . The transition temperature 37.1 K is approximating to that obtained from the susceptibility measurement, 37.9 K. Theoretically, the latent heat should not exist in the second-order transition, however the excess specific heat has an apparent cusp with Δ C _P = 0.88 J/mol K at T _M = 37.1 K. Instead of the simple step classically associated with a second-order transition, the cusp of excess specific heat signifies the contributions from thermal and spin fluctuations. ^{[ 24 ]} According to the analysis of Mozurkewich, ^{[ 25 ]} such fluctuation behavior of C _P near T _M can be divided into two regions: one is the critical region where the critical fluctuations play a crucial role, and the other lies outside the critical region where Gaussian fluctuations are dominant. In the region of Gaussian fluctuation, the specific heat could be described as ^{[ 25 ]}

where the polynomial sum represents a smooth background variation, H is the magnitude of the mean-field step, and G is the strength of the Gaussian term whose form assumes three-dimensional fluctuations. The valid area of Gaussian fluctuations is found as follows. We take a _n , G , and H as the adjustable parameters and fit the C _P curve with formulas ( 6a ) and ( 6b ) in the temperature region ∣ T – T _M ∣ > T * by minimizing the square error σ ² . The T * dependence of G and σ ² is shown in Fig. 5 . When T * ≤0.5 K, with the increase of T *, σ ² drops drastically to the minimum, while G enhances quickly; when T * > 0.5 K, as T * increases, σ ² is almost unchanged and the enhancement of G is comparatively slow, indicating that the data no longer diverge much from the Gaussian prediction. Thus the width of the critical region of ZnV ₂ O ₄ is T * = 0.5 K. In the critical region, the critical behavior is further analyzed by renormalization-group theory, the excess specific heat could be described as ^{[ 26 , 27 , 28 ]}

where t = ( T – T _M )/ T _M is the reduced temperature, A ⁺ and A ⁻ are the critical amplitudes of specific heat above and below the transition temperature, respectively, α is the critical exponent, and F ⁺ and F ⁻ are the amplitudes of the antiferromagnetic empirical correction term. The critical exponent α depends on dimension d and order parameter n of the system. ^{[ 15 ]} Renormalization-group theory classifies such critical behavior into three different universality classes: 3D Ising model, 3D XY model, and 3D Heisenberg model. ^{[ 15 ]} By using formula ( 7 ), we fit the excess specific heat cusp in the critical region, as shown in Fig. 6 . The obtained critical exponent and amplitude ratio are α = −0.011, and A ⁺ / A ⁻ = 1.46, respectively. With these two coefficients, the specific heat curve in the critical region distributes into two separate branches above and below T _M , indicating the authenticity of the critical exponent fitting. By comparing the critical exponent α and the amplitude ratio A ⁺ / A ⁻ with the theoretical values listed in Table 2 , ^{[ 29 – 32 ]} we find that the amplitude ratio inclines to and slightly deviates from that of the 3D Heisenberg model, while the critical exponent α is close to that of the 3D XY model.

Fig. 5.

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	Fig. 5. The T * − dependent strength of the Gaussian term G (hollow orange hexagons) and square error σ 2 (hollow blue stars).

Fig. 6.

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	Fig. 6. Experimental (triangles) and fitted curves (lines) of the specific heat vs. reduced temperature above (pink) and below (navy blue) T M for PM–AFM transition of ZnV 2 O 4 .

Table 2.

Table 2.

Table 2. Critical exponents and critical amplitude of ZnV 2 O 4 ; theoretical values of different models. . Model Critical exponent α Critical amplitude ratio A + / A − 3D Ising 0.11 0.52 3D XY −0.014 1.06 3D Heisenberg −0.115 1.52 This work −0.011 1.4

Table 2. Critical exponents and critical amplitude of ZnV 2 O 4 ; theoretical values of different models. .

To understand the abnormal critical behaviors of ZnV ₂ O ₄ , it is helpful to compare the critical behaviors of ZnV ₂ O ₄ with those of MnV ₂ O ₄ , since they are isostructural counterparts. In MnV ₂ O ₄ , the second-order transition is ferrimagnetic and happens at higher temperature than the structural and orbital transition. So for MnV ₂ O ₄ , around T _M , the orbital order is lacking, and the spin–spin interactions dominate this second-order transition. Nevertheless, the investigation of DC-magnetization by Zhang’s group revealed that magnetic critical exponent β deviates from the 3D Heisenberg model and proposed that the deviation is due to the precursor phenomenon of orbital order fluctuation. ^{[ 16 ]} For ZnV ₂ O ₄ , T _M < T _S , i.e., around T _M of ZnV ₂ O ₄ , the orbital order has already presented. So the spin–spin and spin–orbital interactions coexist at T _M and both of them contribute to the transition. Therefore, strong spin–orbital coupling might confine the spin degree of freedom, and also induces the universality class of critical behaviors to deviate from the 3D Heisenburg model, even shifting to the 3D XY model.

The phase transition and low temperature characteristics of spin–orbital coupling spinel ZnV ₂ O ₄ are investigated. The Debye temperature, the Fermi energy, and the density of states near the Fermi surface of this compound are given by analyzing the data of specific heat below 5 K. The measurement of temperature-dependent specific heat confirms that the structural transition accompanied with orbital order at T _S = 50.1 K is a first-order one, while the AFM transition at T _M = 37.1 K is a second-order phase transition. For the first-order structural and orbital transition, the enthalpy and entropy change are calculated, meanwhile, the derivative of pressure with respect to temperature d P /d T at T _S is estimated by using the Clausius–Clapayron equation. For the second-order AFM transition, the analysis of the critical behavior estimates the width of the critical fluctuation region to be about 0.5 K, and also finds that the critical amplitude ratio deviates from the theoretical value of the 3D Heisenberg model, while the critical exponent α is close to that of 3D XY model. This kind of abnormal critical behavior might be attributed to the influences of spin–orbital coupling.