Hybrid organic-inorganic compounds are an attractive family of materials in condensed matter physics, which have been extensively studied over the past decade for both fundamental science and technological applications.^[1–5] Because of the combination of organic and inorganic moieties within a single structure, a variety of interesting properties have been discovered in these compounds.^[6,7] In some organic-inorganic hybrids, the inorganic atoms are connected through covalent and ionic interactions to form the inorganic network. Meanwhile, the organic blocks can be connected by weaker interactions such as hydrogen and van der Waals bonding. Thus, a layered structure with quasi-two-dimensional magnetic interaction can be realized in them. Recently, two-dimensional (2D) magnetism has been a hot topic. It was previously thought that the surviving of long-range ferromagnetic (FM) order in 2D was hard due to increased fluctuation. However, the magnetic anisotropy in 2D magnetic crystals could stabilize the long-range magnetic order.^[8]

To investigate the 2D magnetic phenomena, a critical exponent survey provides rich information. Accordingly, by evaluating the critical behavior, the nature of magnetic phase transition and inherent characteristics can be determined.^[9]

Since the critical phenomena of magnetic organic-inorganic hybrids have not yet been well studied, in this paper, we performed a study on the critical behavior of a layer-structured organic-inorganic hybrid, (CH₃NH₃)₂CuCl₄. Various methods were used to estimate the critical exponents near the ferromagnetic–paramagnetic phase transition. A crossover from 2D to 3D long-range order was identified based on the analysis of the obtained critical exponents.

Single crystal samples of (CH₃NH₃)₂CuCl₄ were prepared via a solvothermal condition method as reported previously.^[10] The structural properties of the samples were studied by x-ray diffraction using a Cu-Kα radiation source. Direct-current (dc) magnetization was measured by using a Quantum Design magnetic properties measurement system (MPMS) with the magnetic field applied parallel to the metal-halogen layers.

When a magnetic material undergoes a second-order phase transition, the critical behavior around Curie temperature T_C can be characterized by a series of interrelated critical exponents, namely, β, γ, and δ.^[11]

The magnetic equations can be given as

The parameter β (in the region below T_C) is controlled by the temperature dependence of the spontaneous magnetization, and the parameter γ (in the region above T_C) is controlled by the temperature dependence of the initial susceptibility. At T_C, the δ exponent demonstrates the field dependence of the magnetization.

The scaling hypothesis provides another way to determine the critical exponents. The reduced magnetic equation of state that follows the critical region theory can be expressed as

The state described by the above equations will fall on two universal curves: one above T_C and the other below T_C. This is an important criterion for the critical regime; the exponents in the vicinity of T_C confirm the universal properties.

Figure 1(a) displays crystal structure of (CH₃NH₃)₂CuCl₄. The octahedral complexes are sandwiched between organic blocks. It has been deduced that the diffraction peaks of the sample are in accord with the monoclinic perovskite structure with P21/a, the details of which were reported in Ref. [10]. Figure 1(b) displays the temperature dependence of magnetization under field cooled (FC) conditions with different applied magnetic fields in the ab plane of (CH₃NH₃)₂CuCl₄. The sample displays a FM–paramagnetic (PM) transition at T_C ∼ 9 K in 0.1 T magnetic field, and the transition temperature shifts to a higher temperature with increasing magnetic field.

Fig. 1.

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	Fig. 1. (a) The crystal structure of the 2D organic-inorganic hybrid compound. (b) Temperature dependence of FC magnetization with different applied magnetic fields in the ab plane.

To determine the nature of the FM–PM phase transition, we plot the M² vs. H/M curves in Fig. 2(a). According to the criterion set by Banerjee,^[12] the positive slope for the full range of temperatures and fields studied indicates a second-order phase transition. In this sample, the slope is positive and accordingly, the FM–PM phase transition is second order.

Fig. 2.

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	Fig. 2. (a) The Arrott plots of M2 vs. H/M. (b)–(f) The modified Arrott plots with various models.

In order to gain a deeper insight into the mechanism of magnetic interactions between the spins during the magnetic phase transition, the critical exponents need to be analyzed.^[13] We use the modified Arrott plots (MAPs) method based on the Arrott–Noaks state equation to obtain the precise values of the critical exponents and the class of universality to which (CH₃NH₃)₂CuCl₄ could belong. The isotherm magnetization data is transmitted to a sequence of M^1/β = f(μ₀H/M)^1/γ curves.

According to this method, the MAP is analyzed by certain types of possible exponents belonging to the triciritical mean-field model (β=0.25 and γ = 1), 3D-Heisenberg model (β = 0.365 and γ = 1.336), 3D-Ising model (β = 0.325 and γ = 1.24), and mean-field model (β = 0.5 and γ = 1), while the 2D-Ising model (β = 0.125 and γ = 1.75) is also considered,^[11] which are shown in Figs. 2(b)–2(f).

Table 1.

Table 1.

Table 1. Comparison of critical exponents of (CH3NH3)2CuCl4 obtained with different theoretical models: modified Arrott plot (MAP), Kouvel–Fisher (KF), critical isotherm analysis (CI). . β γ δ Mean-field 0.5 1 3 3D-Heisenberg 0.365 1.336 4.8 Triciritical mean-field 0.25 1 5 3D-Ising 0.325 1.24 4.82 2D-Ising 0.125 1.75 15 This work (MAP) 0.22 0.82 4.72 This work (KF) 0.22 0.81 4.68 This work (CI) 4.4

Table 1. Comparison of critical exponents of (CH3NH3)2CuCl4 obtained with different theoretical models: modified Arrott plot (MAP), Kouvel–Fisher (KF), critical isotherm analysis (CI). .

The critical exponents for the above-mentioned models have been predicted by particular values. The mean-field model is suitable to describe the long-range magnetic interaction, which is equivalent to the classic Landau model. The tritical mean-filed model is used to explain the first-order and second-order magnetic interactions of the magnetic phase transition. 3D-Ising is appropriate for understanding the magnetic anisotropic uniaxial magnetocrystalline interaction^[14] and 3D-Heisenberg is suitable for investigating the short-range magnetic interaction. The critical behavior is impressively influenced by the magnetization data in high magnetic field regions due to interplay variables such as charge, lattice, and orbital degree of freedom in a ferromagnetic system.^[15] The lower-field data have not been considered in the fitting process, because they represent the arrangement of magnetic domains.^[16]

As seen in Figs. 2(b)–2(f), these curves display almost straight lines in the high-field regime. It is therefore difficult to identify the best model. In order to solve this problem, we calculate their relative slopes (RS) defined as RS = S(T)/S(T_C), where S(T) and S(T_C) are the MAP slopes at temperatures T and T_C, respectively. The RS of the most appropriate model should be close to the optimal unity value.

Figure 3 shows the RS vs. T plots using the five distinct models. It demonstrates that the system is well defined by the tricritical mean-field model. The chosen exponents have β = 0.25 and γ = 1 as the initial values.

Fig. 3.

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	Fig. 3. Temperature dependence of the normalized relative slope for different models.

By following a standard procedure to identify the exact critical exponents β and γ, the linear extrapolation from the high-field straight line portions of the isotherms to the M^1/β and (μ₀H/M)^1/γ axes provides the spontaneous magnetization M_S(T,0) and the initial susceptibility , respectively. By fitting the data of M_S(T,0) and following Eqs. (1) and (2), a set of β and γ can be obtained. Then, the obtained new critical exponents are used to repeat this procedure until convergence is reached and the values of β and γ are stable. The intersection of the linear extrapolation of the M_S(T) and straight lines gives the value of T_C. The final values of the critical exponents generated are β = 0.22 and γ = 0.82, as depicted in Fig. 4.

Fig. 4.

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	Fig. 4. Temperature dependence of the spontaneous magnetization MS and the inverse susceptibility .

Alternatively, the critical exponents and T_C can be specified more accurately by the Kouvel–Fisher (KF) method based on the following equations:

As seen in Fig. 5, the new exponents of the linear fits are obtained as β = 0.22 with T_C = 15.52 and γ = 0.82 with T_C = 15.93, respectively. It is worthy to note that the values of the critical exponents and T_C deduced from the KF method agree with the estimation from the MAP technique.

Fig. 5.

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	Fig. 5. The KF plots of MS and .

The third critical exponent δ can be calculated directly from the fitting of the critical isotherm M vs. H at T_C (Fig. 6(a)). According to Eq. (3), the high-field region of the critical isotherm at T = 16 K in log–log plot would give a solid straight line with a slope of 1/δ (Fig. 6(b)). Subsequently, the δ exponent is obtained as 4.4.

Fig. 6.

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	Fig. 6. (a) Isothermal M(h) curve at TC. (b) The same plot on a log–log scale. The straight line is the linear fit of Eq. (3).

According to Wisdom’s scaling relation, the exponent δ can also be theoretically verified by the following relationship:

Since the obtained exponents of the sample are not matched with conventional universal classes, we investigate whether the acquired values of the critical exponents can generate the equation of the Widom’s scaling or not.^[15,17]

The scaling analysis of Eq. (4) can be used in the critical region to further confirm the validity of the results obtained so far and is calculated with the obtained critical exponents. The scaling equation implies that M/ε^β vs. H/ε^{β + γ} yields two universal curves for T > T_C and T < T_C, respectively.

Figure 7(a) shows that the experimental data collapse into two separate branches, one below T_C and the other above T_C. Figure 7(b) illustrates the same plots on the log–log scale. The experimental data collapse in the higher-field region into two distinct curves. This supports the validity of the obtained values of the critical exponents and T_C. However, while the scaling is effective in the high magnetic field regime, a tiny divergence is present in the low magnetic field regime. This can be attributed to the magnetic inhomogeneity. FM clusters may generate a local anisotropic field. When an applied field is not sufficiently high to overcome the local anisotropic field, the magnetic moment can be aligned with a local anisotropic field. As a result, some magnetization data points do not entirely fall on the branches.^[18,19]

Fig. 7.

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	Fig. 7. (a) Scaling plots of renormalized magnetization M| ε |−β vs. H| ε | −(β + γ) using β and γ determined by the KF method. (b) The alternative plot on a log–log scale.

The Arrott–Noakes equation of state is

Fig. 8.

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	Fig. 8. (a) The (H/M)1/γ vs. (M)1/β curves with the critical exponents obtained from the KF method. (b) Temperature dependence of modified coefficients A′ and B′.

The obtained critical exponents in the present study do not belong to any predicted class of universality. The effective critical exponents (β_eff and γ_eff) have been examined as a function of ε using Eq. (9) in order to determine whether they match any universality class in the asymptotic area of the system,

Fig. 9.

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	Fig. 9. Effective critical exponents (a) βeff and (b) γeff as a function of the reduced temperature ε.

The origin of the FM interaction in this compound is due to the magnetic ions (Cu) in the ab plane and the antiferro distortive displacement of the halide ions within the ab plane, which is the consequence of Jahn–Teller distortion of the octahedral complex in the inorganic sheet. As a result, the single unpaired electron is located in the d_x²–y² orbital at each of the 3D holes of the Cu²⁺ ions in this compound, which aligns with the long axis of the octahedron and the spins experience FM superexchange, leading to a long-range FM order.^[2,24–26]

It is clear from the previous sections that the FM interaction in this system does not completely conform to any predictive model. The results from the report of Taroni et al. show that the critical exponent β for the bidimensional system should be within the range 0.1 ≤ β ≤ 0.25.^[26] The spontaneous survey magnetization by Jongh et al. illustrates that the critical exponent β is approximately 0.125 for the 2D system, except in the region very close to the ordering temperature; β will be changed to the 3D value of approximately 0.33 according to Griffiths universality hypothesis.^[27] The β value of (CH₃NH³)₂CuCl₄ is close to 0.25; this value lies between those of the 2D and 3D characteristics, which indicates that the interlayer coupling should be considerable. The exchange in the layer and between layers is typically strong and weak, respectively. Namely, the intralayer exchange is FM while the interlayer coupling is antiferromagnetic. For this sort of material including anisotropic intralayer exchange, a 2D-type behavior is predicted, but in a narrow ordering temperature (near T_C), the crossover from 2D to 3D caused by the interlayer interaction begins to take effect.^[28] The γ value is close to the mean-field model prediction and δ is located between the long-range and short-range couplings. These exponents can be attributed to a crossover effect of the 2D to the 3D system. To describe the crossover effect, we should consider the interlayer coupling.

The results indicate that the critical exponents in this compound do not strictly belong to the common universality models. Accordingly, it is important to determine the nature and the range of the interaction in the present study. For a homogeneous magnetic system, the universality of the magnetic phase transition is strongly dependent on the exchange interaction J(r). Renormalization group theory treats the magnetic ordering as an attractive interaction of spin which decays with distance r as

It has been argued that if σ ⩾ 2, the 3D-Heisenberg parameter is valid, where J(r) decreases faster than r⁻⁵. When σ ⩽ 3/2, the conditions for the mean-field model are satisfied so that J(r) decreases more slowly than r^−4.5. In this research work, by using Eq. (12) we see that σ = 122 (d = 3, n = 3), so the exchange distance decays as J(r)≈ r^−4.22, in agreement with a long-range interaction.

The previous study of the field dependent susceptibility demonstrated a very weak FM coupling near T_C between the layers that causes the space dimensionality of the system to crossover to a 3D long-range order.^[25] From the studies performed so far, it can be concluded that the strong magnetic anisotropy, as well as the dipole interaction and inevitable interlayer interaction, can induce the system to display a 3D long-range magnetic characteristic above T_C.^[29]

We have provided a detailed analysis on the critical exponents of the organic-inorganic hybrid with the formula (CH₃NH₃)₂CuCl₄. The system undergoes a second-order PM–FM phase transition. By using various methods, such as MAPs, Kouvel–Fisher, and critical isotherm analysis, the reliable critical exponents of β = 0.22, γ = 0.82, and δ = 4.4 were estimated. The critical exponents of this system do not belong to any universality class. The value of β is between those of the 2D and 3D systems and suggests a crossover from 2D to 3D near the Curie temperature.