Critical exponent analysis of lightly germanium-doped La_0.7Ca_0.3Mn_1-xGe_xO₃ (x = 0.05 and x = 0.07)

Modern society has developed refrigeration and uses it daily. Refrigeration has become a significant part of the food distribution system, and the food industry employs both chilling and freezing to slow the physical, microbiological, and chemical activities that cause food deterioration. Food is cooled from ambient temperature to above 0 $°$C when chilled and to between −18 $°$C (255 K) and −35 $°$C (238 K) when frozen.¹ Unfortunately, conventional mechanical refrigeration technologies contribute significantly to the environmental impacts of the food sector via both direct and indirect greenhouse gas emissions. To reduce these environmental emission impacts, worldwide research and development efforts have been aimed at improving refrigeration performance and developing new refrigeration technologies with the potential for reduced environmental impact.¹ Magnetic refrigeration is among the cooling technologies with a high level of current research activity. Unfortunately, implementation is not imminent, as better magnetocaloric materials and other fundamental breakthroughs are needed to achieve alternative cooling and near-room-temperature refrigeration technologies.² 

Recently, Tagliafico et al. reported a linear reciprocating magnetic refrigerator prototype that achieves a maximum temperature difference of 5.0 K. Their technology operates near room temperature, under free-run conditions. Their active material includes gadolinium plates, and their process is based on the active magnetic regenerative thermodynamics cycle.³ However, the commercial price of pure gadolinium metal is $450 per 100 g. Therefore, alternative materials based on manganite perovskites have been proposed. In contrast to gadolinium, pure manganese costs $6.5 per 100 g. One can find a brief overview of the magnetocaloric effects associated with manganite perovskites in a study by Phan et al.⁴ 

The Curie temperature, T_C, is important to the characterization of magnetic cooling materials. It is the point at which a ferromagnetic material becomes paramagnetic. Thus, the phase transition from a paramagnetic (PM) insulator to ferromagnetic (FM) should be analyzed via a critical exponent study designed around the Curie temperature. Ahmed et al. reported that the T_C values of materials in the La_0.7Ca_0.3Mn_1-xGe_xO₃ (x = 0.02, 0.03, 0.04, 0.06 and 0.08) series first decrease and then increase with x. The Curie temperatures recorded were -74.2 ^oC, -85.2 ^oC, -40.1 ^oC, -50.8 ^oC, and -108.7 ^oC, respectively.⁵ This T_C fluctuation is recurrent behavior of magnetisation and resistivity due to formation of Ge pairs next to Mn⁺³/Mn⁺⁴ pairs that can be explained by Zener bond blocking via single site Ge-substitution in La_0.7Ca_0.3MnO.⁶ In addition, Ge doping provides an extreme example of carrier trapping due to formation of magnetic or lattice polarons. In the former case, carriers are confined in a finite region, though not necessarily an isolated lattice site, and the trapping centers can be thermally active.^7–10 A systematic difference between Mn³⁺ and Mn⁴⁺ substitution might be expected in B-site doping, which reduces the mobile electron content directly, e.g., when Mn⁴⁺ or Mn³⁺ (hole) is replaced. This mechanism tends to distort the local lattice by forming a Jahn–Teller polaron, while a hole does not. The concentrations of the two carriers can vary in manganese oxides.⁷ In this work, we report a critical behavior analysis of polycrystalline La_0.7Ca_0.3Mn_1-xGe_xO₃ (x = 0.05 and x = 0.07). The critical parameters of 5- and 7-percent Ge-doped LCMO were determined to be T_C = 185 K, β = 0.331 ± 0.019, and γ = 1.15 ± 0.017; and T_C = 153 K, β = 0.496 ± 0.011, and γ = 1.03 ± 0.046, respectively, via the modified Arrott plot method. Our work may contribute further to the work by Ahmed et al., in which the T_C values of lightly Ge-doped La_0.7Ca_0.3Mn_1-xGe_xO₃ vary non-linearly.⁵ 

Polycrystalline La_0.7Ca_0.3Mn_1-xGe_xO₃, (x = 0.05 and 0.07) was synthesized from high-purity La₂O₃, CaCO₃ MnCO₃, and GeO₂ via a conventional solid-state reaction technique which the same technique was used by ref. 5. The sample was prepared at 1000 ^oC for 12 h, and then ground, pressed, and finally sintered at 1200 ^oC for 24 h. X-ray diffraction (XRD) patterns of all samples were measured using a Brucker AXS D8 Discover system with Cu K–α radiation at room temperature. The magnetic properties of the compound were characterized by using a vibrating sample magnetometer (VSM).

The crystal structure of La_0.7Ca_0.3Mn_1-xGe_xO₃ (x = 0.05 and x = 0.07) was investigated via X-ray diffraction. The lattice parameters obtained are a = 5.46 Å, b = 7.73 Å, c = 5.48 Å, and volume = 231 ų; and a = 5.45 Å, b = 7.69 Å, c = 5.47 Å, and volume = 229 ų for x = 0.5 and x = 0.07, respectively. The XRD patterns indicate clean, single-phase orthorhombic polycrystalline structures with space group Pbnm, as depicted in Fig. 1. The lattice parameters of the Ge-doped samples do not differ substantially.

We also collected magnetothermal data to determine whether the magnetization versus temperature curve exhibits a gradual FM to PM transition or drops sharply. A continuous FM to PM transition is associated with a second-order magnetic phase transition (SOMT), while a sharp drop is associated with a first-order magnetic phase transition (FOMT). In the present work, both LCMO samples exhibit gradual FM to PM transitions as the temperature increases under an external field of 100 Oe (figure not shown). We also investigated isothermal magnetization of both samples around the phase transition temperature (T_C). Using the isothermal data, we obtained the slope of the H/M versus M² curve. The Banerjee criterion can confirm that the sample exhibits a SOMT if the slope is positive or a FOMT if the slope is negative.¹¹ Both samples have positive slopes and thus exhibit SOMTs, as shown in Fig. 2.

According to the Arrott technique, a plot of H/M versus M² should exhibit parallel linear curves in high fields near T_C. This is based on mean-field theory and critical exponent values of β = 0.5 and γ = 1. Mean-field theory is intertwined with long-range FM order. If the plot of H/M versus M² is not nonlinear-parallelized, then mean-field theory cannot be used to describe magnetism in the present samples. Therefore, it is important to determine the exponential parameters β and γ of the samples.

The critical exponents β and γ obtained via fitting are associated with spontaneous magnetization and initial magnetic susceptibility, which can be used to describe a system of magnetic interactions theoretically. The β and γ values can be determined using a modified Arrott method by plotting (H/M)^1/γ versus M^1/β, where trivial critical exponents of β=0.5 and γ=1 (from the mean-field theory model) are used as starting points. Then, one might measure spontaneous magnetization versus temperature in the high-magnetic field region based upon the intersection of the straight extrapolation line with the axis. Similarly, the relationship between initial magnetic susceptibility and temperature is determined inversely from the intersection with the axis. Near the T_C, M_S(T) and χ₀^-1 (T) are described by the following power law dependence relations:¹² 

where M₀, h₀, and D are the critical amplitudes, and ε = (T – T_C)/T_C is the reduced temperature. According to Eq. (1) the slope obtained from the straight-line extrapolation fitting of log [M_S (T)] versus log (ε) produces a new value for β. Similarly, Eq. (2) indicates that the straight-line extrapolation fitting of log [(χ₀^-1(T)] versus log (ε) produces a new γ. Thus, the free parameter T_C in Eq. (1) and (2) is adjusted to yield the best fit while also fitting the straight-line extrapolation system.

The Arrott plot is nonlinear and exhibits upward curvature even in the high-field regime, indicating that a case where β = 0.5 and γ = 1 does not satisfy the Arrott-Noakes equation of state (H/M)^(1/γ) = (T–T_C)/T_C + (M/M₁)^1/β.¹³ Thus, the Landau phase transition and the mean field theories based upon β = 0.5 and γ = 1 are not valid for the La_0.7Ca_0.3Mn_0.095Ge_0.05xO₃ and La_0.7Ca_0.3Mn_0.093Ge_0.07O₃ systems. Therefore, modified Arrott plots were used to obtain the proper β and γ values from the following equation of state:

where a and b are constants. If β and γ are selected properly, the transform of the isothermal curve becomes a set of close, parallel straight lines in the high-field region. To determine the correct β and γ, we tested trial values of β and γ and evaluated their applicability to the mean-field theory model. These values of β and γ were substituted into Eq. (4) to generate graphs similar to those in Fig. 4 (not shown). In the high-field region, we perform linear extrapolation of the isothermal data to give (M_S)^1/β and (χ₀^-1)^1/γ as intercepts on the (M)^1/β and (H/M)^1/γ axes, respectively. The new values of β and γ are repeatedly used to reconstruct new modified Arrott plots until stable values of, β, γ, and T_C are obtained. After this routine, a set of reasonably good straight, parallel lines are generated alongside several parameters: β = 0.331 ± 0.019 315 and γ = 1.15 ± 0.017 for x = 0.05; β = 0.496 ± 0.011 and γ = 1.03 ± 0.046 for x = 0.07. Fig. 3 shows the final β and γ values based on M_S (T) and χ₀^-1(T) as functions of temperature. We shall use average values (T_C = 185 K for x = 0.05 and T_C = = 153 K for x = 0.07) for the following calculations and discussion.

The magnetic equation of state can be expressed using the variables M (H,ε), H, and T. The former variable can be defined via the scaling hypothesis:

where f₊ for T > T_C and f_- for T < T_C are regular analytic functions. After renormalization of Eq. (5), it appears that for the correct scaling relations and proper choice of β and γ values, the scaled function falls on two universal group curves, with some curves above T_C and other curves below T_C. The Landau theory of phase transition expresses the Gibbs free energy G as a function of the order parameter M:

where coefficients a and b are temperature-dependent.¹⁴ Upon applying an equilibrium condition (dG/dM = 0, i.e., energy minimum), the magnetic equation of state becomes:

Therefore, H/M versus M² appears linear in the high-field region of the Arrott plot. Eq. 7 is yielded from derivate of Eq. 6 then Eq. 7 can be used to produce modified Arrott plots at various temperatures. The intercept of H/M versus M² on the H/M axis is positive/negative above/below the T_C. The H/M versus M² lines that indicate T_C should cross the origin as plotted in Fig. 4.

The resulting β and γ values can be examined via scaling theory. Eq. 5 implies that M|ε|^β versus H/|ε|^(β+γ) should produce two universal curves: one set for temperatures below T_C, and another for those above T_C. Scaled data for the x = 0.05 and x = 0.07 samples is shown in Figs. 5a and 5b, respectively, based on the final values of β and γ determined above. One can observe that all of the high-field data points fall on two individual branches of T_C>T and T<T_C. This means that the La_0.7Ca_0.3Mn_1-xGe_xO₃ samples undergo SOMT. However, scaling describes most high fields, although it is poor in low-magnetic fields because of the rearrangement of magnetic domains and the effect of uncertainty in calculating the demagnetization factor, which can influence this region significantly. The separate branches indicate clearly that the interaction is suitably renormalized in the critical region using the scaling equation of state.

These results prove that FM interactions in La_0.7Ca_0.3Mn_1-xGe_xO₃ (x = 0.05 and x = 0.07) have the following properties: (i) they follow a short-range interaction model in which the estimated exponents are between the values expected for the 3D Heisenberg and 3D Ising models (ii) they are long-range interactions because the calculated exponents are near those associated with the mean-field model. The final β values of the x = 0.05 sample are between those expected for the 3D Heisenberg and 3D Ising models, and become close to those required for the mean-field model upon increased Ge doping, as shown by the x = 0.07 sample. It is thought that lightly Ge-doped samples in which Ge⁴⁺ ions replace the Mn⁴⁺ ions are not robust enough to influence the colossal magnetic ordering from ferromagnetic to paramagnetic, as would be indicated by a first-order magnetic phase transition of La_0.7Ca_0.3MnO₃. Increasing the Ge doping concentration weakens the Jahn-Teller effect and suppresses short-range magnetic interactions. According to Millis polaron theory,¹⁵ the Jahn-Teller effect is assumed from the electron-phonon coupling and lattice distortions that split the on-site orbital degeneracy of the levels/symmetry distortions of the oxygen octahedra around a Mn site. Increasing the substitution of Mn⁴⁺ with Ge⁴⁺ might decrease the Mn³⁺/Mn⁴⁺ ratio and electron-phonon coupling strength.

We successfully synthesized La_0.7Ca_0.3Mn_1-xGe_xO₃ (x = 0.05 and x = 0.07) via solid-state reactions and studied their magnetic interactions via critical exponent analysis. Doping of Ge onto the Mn sites in La_0.7Ca_0.3MnO₃ might be evidence of direct substitution of Mn⁴⁺ that causes magnetic ordering to transform from short-range to long-range, as described by the β and γ values obtained. Thus, Ge doping might also weaken electron-phonon coupling.

This work was primarily supported by Chungbuk National University, South Korea. DN received partial support for this work from Syarif Hidayatullah State Islamic University via an international collaborative research grant (Un.01/KPA/568/2017). BK was partially supported by a grant from the Ministry of Research, Technology and Higher Education, the Republic of Indonesia Government via Universitas Indonesia (contract for PUPT grant no. 2699/UN.2.R3.1/HKP.05.00/2017).