Studies on the magnetoelastic and magnetocaloric properties of Yb_1−xMg_xMnO₃ using neutron diffraction and magnetization measurements

Abstract

We report the magnetic ordering and magnetoelastic coupling of polycrystalline hexagonal Yb_1−xMg_xMnO₃ (x = 0.00 and 0.05) compounds by using neutron diffraction measurements. The magnetocaloric properties of these Yb_1−xMg_xMnO₃ compounds are also studied using magnetization measurements. The temperature dependence of the lattice parameters (a and c/a ratio) and unit cell volume V show anomalous behavior near T_N1 ∼ 85 K (the Mn ordering temperature) due to the magnetoelastic effect. Also all the Mn–O bond distances display considerable variation at T_N1. Isothermal magnetization curves measured near the Yb long range ordering temperatures indicate a field induced magnetic transition with applied field. The isothermal magnetic entropy change (−ΔS_M) is calculated from the magnetization curves measured for different temperatures. Values of maximum entropy change (−ΔS^max_M), the adiabatic temperature change (ΔT_ad) and the relative cooling power (RCP) for these compounds are found to be 3.02 ± 0.37 J mol⁻¹ K⁻¹, 8.6 ± 0.95 K and 41 ± 9 J mol⁻¹ for x = 0.00, and 2.63 ± 0.36 J mol⁻¹ K⁻¹, 9.06 ± 0.96 K and 40.0 ± 10 J mol⁻¹ for x = 0.05, respectively, for ΔH = 100 kOe. Rescaling of the −ΔS_M vs. T curves for various fields fit into a single curve, implying the second-order phase transition.

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1. Introduction

The hexagonal rare-earth manganites, RMnO₃ (R = Ho, Er, Tm, Yb, Lu, Y etc.) show magnetoelectric coupling effects in which both ferroelectricity and magnetic ordering can exist simultaneously in the ordered phase. Such materials, called multiferroics, have potential for a variety of applications since the magnetic ordering can be controlled by an electric field and vice versa.^1,2 Besides the magnetoelectric coupling, there are reports on these materials^3–5 showing coupling between magnetic and lattice degrees of freedom giving rise to interesting magnetoelastic effects.

Hexagonal YbMnO₃ consists of layers of corner-sharing MnO₅ trigonal bipyramids separating layers of Yb³⁺ ions. In the MnO₅ trigonal bipyramids, each manganese ion is surrounded by three in-plane and two apical oxygen ions. A net electric polarization arises in YbMnO₃ due to buckling of the layered MnO₅ polyhedra accompanied by displacements of the Yb ions resulting in ferroelectric behavior with ferroelectric transition at 990 K.⁶ YbMnO₃ has a complex magnetic phase diagram below the Neel temperature (∼85 K), the magnetic structure of YbMnO₃ can be described as frustrated antiferromagnetic on a triangular lattice in the ab-plane.⁷ The Mn³⁺ moments get aligned in the ab-plane to produce antiferromagnetic order at the Néel temperature (T_N1 = 85 K) while the Yb moments at the 4b crystallographic site order due to the Mn molecular field.⁸ Further, Yb³⁺ moments at the 2a crystallographic site show long range ordering through Yb–Yb interactions below T_N2 ∼ 3.5 K.^8,9 It is reported, a magnetic field applied along the c-axis below T_N2 ∼ 3.5 K induces magnetic transition at a critical field of H||c ∼ 35 kOe (ref. 9) due to the spin-flip/reorientation of the Yb(2a) moments.⁸

The interesting behavior of YbMnO₃ motivated us to study the magnetic structure and spin lattice coupling of YbMnO₃ and Yb_0.95Mg_0.05MnO₃ compounds using neutron diffraction. The later compound was studied to study changes due to Mg substitution. In our previous studies,^7,10 we reported room temperature neutron-diffraction (ND) results of these samples presenting structural parameters like cell volume, bond lengths and angles at room temperature (300 K). Our previous studies based on temperature depended magnetization and heat capacity^7,10 show Yb³⁺ ordering at around 3.5 K, and antiferromagnetic Mn³⁺ ordering around 85 K. Here, in the present work, we extend the neutron-diffraction (ND) studies to lower temperatures, presenting the temperature dependence of various structural parameters near the magnetic transition/ordering temperatures. In the earlier reports^10,11 the magnetocaloric properties were studied using heat capacity measurements. In the present work we also present results on the isothermal magnetization measurements carried out around the Yb³⁺ ordering temperature and evaluate magnetocaloric properties from the isothermal magnetization data. The magnetocaloric properties evaluated from the isothermal magnetization data in the present work is complementary to the magnetocaloric properties evaluated from the heat capacity measurements in.^10,11

2. Experimental details

Polycrystalline samples of Yb_1−xMg_xMnO₃ (x = 0.0 and 0.05) were synthesized by a conventional solid-state reaction method from stoichiometric mixtures of Yb₂O₃, MgO, and MnCO₃ powders. The experimental details of sintering, measurement of X-ray diffraction (XRD) patterns, magnetic and thermal properties are given in our previous papers.^7,10 In addition, the neutron-diffraction (ND) measurements were carried out on powder samples with a wavelength of 1.48 Å using a multi-position sensitive detector based focusing crystal diffractometer PD-3 set up by the UGC-DAE Consortium for Scientific Research at the National Facility for Neutron Beam Research NFNBR, Dhruva reactor, Mumbai, India.¹² Neutron diffraction (ND) patterns were recorded at different temperatures between 2.5 K and 300 K. The magnetization measurements were performed using a Cryogenic Inc. (UK) make vibrating sample magnetometer operating at 20.4 Hz. Magnetization isotherms were measured at different temperatures in fields up to 100 kOe near the low temperature ordering transition.

3. Results and discussions

3.1. Neutron diffraction

Neutron diffraction (ND) measurements on polycrystalline Yb_1−xMg_xMnO₃ (x = 0.00 and 0.05) were performed at different temperatures near the Mn ordering temperature with incident neutron wavelength λ = 1.48 Å. Room temperature ND results on these compounds reported earlier⁷ showed subtle structural changes in cell volume, bond lengths and angles and polyhedral distortions. The nuclear structure peaks can be described by a single phase with space group P6₃cm at all temperatures. The temperature variation of the ND patterns is shown in Fig. 1a. The figure shows that the intensity of (102) as well as (101) Bragg positions increases below T_N1 ∼ 85 K and the intensity of (100) increases below T_N2 ∼ 3.5 K, where the nuclear reflections are forbidden by the P6₃cm symmetry. The temperature variations of the integrated intensities of Bragg peaks (100), (101), and (102) are shown in Fig. 1b. It is observed that large changes occur in the integrated intensities of these Bragg peaks around antiferromagnetic transition T_N1 ∼ 85 K and ferromagnetic transition T_N2 ∼ 3.5 K which correspond to the long range magnetic ordering of Mn and Yb ions, respectively. It is further observed that the intensity at the onset/start of corresponding Bragg peaks intensity slightly shift to higher temperature for x = 0.05 sample. The (100) and (101) Bragg peaks differ from each other with temperature dependences. The (101) peak intensity rises abruptly below 85 K while the (100) peak intensity remains rather weak down to about 10 K then suddenly increases below 10 K. The observed temperature dependences suggest that the (101) and (100) peak intensities are controlled by the Mn and Yb orderings, respectively, while the (102) Bragg peak probably has contributions from both orderings.⁸ The observed temperature dependences of the (101) peak in YbMnO₃ is similar to that of the (100) peak in YMnO₃,^13,14 whereas the Mn order does not induce a strong (101) magnetic peak in YMnO₃ which suggests that the Mn order is affected by the nature of the R ion.

Fig. 1 (a) Neutron diffraction patterns (b) integrated intensities of the (100), (101), and (102) reflections between 2.8 K and 100 K of Yb_1−xMg_xMnO₃ (x = 0.0 and 0.05).

Fig. 2 shows Rietveld refinement of temperature dependent ND data for selected temperatures (2.5 K, 10 K and 70 K) which confirm the hexagonal structure of Yb_1−xMg_xMnO₃ with space group P6₃cm. To analyze these results, one need to remember that six different irreducible representations are possible with k = 0 according to magnetic group theories.¹³ Only four unidimensional irreducible representations seem to be particularly favorable among all the hexagonal manganites known to date. For example, the magnetic peaks of YMnO₃ can be explained by either Γ₁ or Γ₃ representation while those of ErMnO₃ are compatible with Γ₂ or Γ₄ representation. Fabreges et al.⁸ reported that the best fit of the YbMnO₃ data was obtained assuming Γ₄ representation for Mn ordering, and the ordered moment at 1.5 K was found to be 3.25 μ_B. For the Γ₁ and Γ₄ representations, the magnetic moment of Mn at the (x, 0, 0) position is aligned perpendicular to the crystallographic a and b axes. Furthermore, the Γ₁ representation has an antiferromagnetic coupling between the moments at z = 0 and z = 1/2 planes whereas for the Γ₄ representation the inter-plane coupling is ferromagnetic. On the other hand, for the Γ₂ and Γ₃ representations, the in-plane magnetic moment of Mn is aligned parallel to the crystallographic a and b axes. For the Γ₂ representation the inter-plane moments are coupled antiferromagnetically, while for the Γ₃ representation the inter-plane moments are coupled ferromagnetically along the c axis.¹⁴ For the magnetic refinement of the present ND data, the structure described by the basis vectors of the irreducible representations Γ₆ for Yb³⁺ (at Wyckoff site 2a), Γ₂ for Yb³⁺ at 4b site and Γ₂ representation for Mn at 6c site were used.⁸

Fig. 2 Rietveld refinement of temperature dependent ND data of Yb_1−xMg_xMnO₃ (x = 0.00 and 0.05) at 2.5 K, 10 K and 70 K.

From the Rietveld refinement of temperature dependent ND data, the structural and magnetic properties of the samples were extracted and selected data are shown in Fig. 3–5. Fig. 3a–c show the temperature dependence of the lattice parameters a, c/a, and the unit cell volume V of Yb_1−xMg_xMnO₃. In general, the lattice parameter a and unit cell volume V decrease and c/a increases with decreasing temperature except near the magnetic transition temperatures (T_N1 ≈ 85 K, T_N2 ≈ 3.5 K for x = 0.00 and T_N1 ≈ 89 K, T_N2 ≈ 3.5 K for x = 0.05). Close to the magnetic transition temperatures all the lattice parameters (a, c/a, and V) show anomalous variation^4,5,15 as a sudden change in value/slope from the background. This anomaly in lattice parameters near the magnetic transition temperatures implies strong spin-lattice coupling and consequently a magnetoelastic coupling.^4,16,17 The magnetoelastic effect is clearly visible in the temperature variation of the unit cell volume V data as shown in Fig. 3d. This figure shows the temperature dependence of experimentally measured cell volume (red square symbol + line) and the calculated unit cell volume (blue solid line) in the absence of the magnetic transition for the Yb_0.95Mg_0.05MnO₃ sample. The temperature dependence of the cell volume in the absence of magnetic transition is obtained using the Debey–Gruneisen approximation.¹⁸ The measured unit cell volume V in the paramagnetic state (T > T_N1) is fitted to the Debey–Gruneisen equation and using the fit parameters (from the high temperature region fitting) the cell volume for the low temperature (T < T_N1) nonmagnetic phase is calculated and is shown (blue line in the figure) in Fig. 3d. By subtracting the calculated volume for the nonmagnetic phase from the measured data, we determined the volume strain ΔV which is plotted in the inset of Fig. 3d. The maximum volume magnetostriction (ΔV/V) near the magnetic transition temperature (T_N1) is found to be ∼0.09% for the Yb_0.95Mg_0.05MnO₃.

Fig. 3 Temperature dependence of lattice parameters for YbMnO₃ and Yb_0.95Mg_0.05MnO₃ (a) lattice parameter a, (b) c/a ratio and (c) unit cell volume V. (d) Temperature dependence unit cell volume data for Yb_0.95Mg_0.05MnO₃ (red square + line) fitted to Debye–Gruneisen equation (blue line). Inset of (d) shows the temperature dependence of the lattice strain ΔV obtained by subtracting the background (fit curve) from the measured V.

Variation of different Mn–O bond lengths for both YbMnO₃ and Yb_0.95Mg_0.05Mn0₃ are shown in Fig. 4a–c as a function of temperature. The Mn–O4 and Mn–O3 bond distances show significant variation at T_N1. The Mn–O4 bond length shows decrease below T_N1 while the Mn–O3 bond length shows increase below T_N1. The Mn–O1 and Mn–O2 bond lengths do not show significant changes with temperature. The difference between Mn–O3 and Mn–O4 bond lengths (ΔL), which reflects the change of the average Mn position along the x direction from its neutral position, varies with temperature as shown in Fig. 4d. Fig. 4e, f and g show the variation of Mn x position, O3 z position and O4 z position in the unit cell as a function of temperature for both YbMnO₃ and Yb_0.95Mg_0.05Mn0₃. The figures clearly show the systematic variations in the atomic positions as the sample cools through T_N1. In Fig. 4h–j temperature dependence of the polyhedral distortion (Δ), the tilting angle and the buckling angle are shown for the samples showing clear signatures at the magnetic transition temperatures. These results of variation of temperature dependence of different structural parameters near the magnetic transition temperature clearly indicate strong magnetoelastic coupling in the YbMnO₃ system.

Fig. 4 Temperature dependence of various structural parameters for YbMnO₃ and Yb_0.95Mg_0.05MnO₃: left panel (a) Mn–O4 bond length, (b) Mn–O3 bond length, (c) the average apical bond length (〈Mn–O1, Mn–O2〉), (d) ΔL, difference between Mn–O3 and Mn–O4 bond lengths and (e) Mn_x atomic position in the unit cell. Right panel: (f),(g) O3 z and O4 z atomic positions in the unit cell. (h) Polyhedral distortion (Δ), (i) tilting angle and (j) the buckling angle for the MnO₅ polyhedra.

Fig. 5a shows the magnetic structure of YbMnO₃ from the Rietveld refinement. Fig. 5b shows the temperature dependence of the ordered Mn and Yb magnetic moments extracted from the neutron diffraction data refinement. Below T_N1, for both compounds, ordered Mn moment increases with decrease in temperature. The value of the saturated moment is 3.55 μ_B for both the samples which is smaller than 4 μ_B for Mn³⁺. In Fig. 5b symbols show the data points and the solid lines show the fit to the data using a self-consistent mean-field calculation described in ref. 8. Following ref. 8, we used the expression to fit the observed moment versus T for both the samples. Here B₂(x) is the Brillouin function for S = 2 and λ₀ is the molecular field constant representing Mn–Mn exchange.⁸ Best fit to the experimental data points are obtained with m_sat = 3.55 μ_B and λ₀ = 20 for undoped YbMnO₃ and with m_sat = 3.55 μ_B and λ₀ = 26 for Mg doped YbMnO₃. These results suggests that with Mg doping at the Yb site, the geometrical frustration for the antiferromagnetic interaction for the triangular lattice of Mn³⁺ ions is reduced and the exchange interaction is strengthened. Compared with the undoped YbMnO₃, higher value of T_N1 (89 K for x = 0.05 and 85 K for x = 0.00), reduced unit cell volume (shown in Fig. 3c) and reduced frustration parameter^7,10 (2.045 for x = 0.05 and 2.576 for x = 0.00) for the Mg doped sample support this fact. The Yb moment values at 2.8 K are found to be m_Yb(4b) = 1.404 μ_B, m_Yb(2a) = 1.4697 μ_B for x = 0.00 and m_Yb(4b) = 1.307 μ_B, m_Yb(2a) = 1.387 μ_B for x = 0.05.

Fig. 5 (a) Magnetic structure of YbMnO₃ form the Rietveld refinement (b). Temperature dependent ordered magnetic moment of Mn for Yb_1−xMg_xMnO₃ (x = 0.00 and x = 0.05). Symbols (open for x = 0.00 and closed for x = 0.05) represent data points and solid lines represent fit to mean-field equation.

3.2. Magnetocaloric effect

We recently reported magnetization results of Yb_1−xMg_xMnO₃ (x = 0.00 and x = 0.05) which show ferromagnetic ordering in Yb³⁺ sub-lattice at low temperature around 3 K, and antiferromagnetic ordering of Mn³⁺ around 85 K.⁷ In Fig. 6a and b show magnetization isotherms, M(H), for Yb_1−xMg_xMnO₃ (x = 0.00 and x = 0.05) taken over a certain temperature range close to their low temperature transition T_N2 at which the Yb ordering takes place. The magnetic field was varied from 0 to 100 kOe. Some representative plots of field variation of M in the temperature range 2–40 K are shown in Fig. 6a and b. From these plots, one can clearly differentiate the nature of H dependence of M below T_N2 from that above T_N2. Below T_N2, M increases slowly with H in the low field region and suddenly the slope changes at a critical field H_c and then increases slowly with further increase of H. This change of slope in M vs. H suggests a field induced magnetic transition.

Fig. 6 Field dependence of isothermal magnetization for Yb_1−xMg_xMnO₃. (a) x = 0.00 and (b) x = 0.05. The Arrott plots of H/M vs. M² at various temperatures for Yb_1−xMg_xMnO₃ (c) x = 0.00 and (d); x = 0.05 samples.

To understand the magnetic phase transition, we have transformed M(H) data into Arrott plots, H/M versus M² as shown in Fig. 6c and d.¹⁹ A positive or negative slope of the Arrot plots (H/M versus M²) indicates a second order or first order transition, respectively.²⁰ Data of both samples show positive slopes in the complete M² ranges indicating that the system exhibits a second-order phase transition.

From the isothermal magnetization curves at different temperatures magnetic-entropy change ΔS_M can be obtained from Maxwell's thermodynamic relationship:²¹

The magnetic-entropy change ΔS_M due to the variation of the applied magnetic field from 0 to H is given by

The value of ΔS_M is determined using eqn (2.3b) and the error/uncertainity in ΔS_M is calculated using eqn (2.4b) of ref. 22. Calculation details of the same are explained in ref. 23. Using the given manufacturer data and following ref. 22, errors of 0.5%, 0.1% and 0.25% were considered for uncertainties in magnetization, magnetic field and temperature values to calculate the error in ΔS_M. The relative error in ΔS_M is 10–20% for ΔH = 100 kOe and T around 10 K.

Values of −ΔS_M for different ΔH as a function of temperature determined under an applied magnetic field up to 100 kOe are presented in Fig. 7a and b. The −ΔS_M is positive in the entire temperature range for both samples. The curves present a characteristic shape with a broad maximum in the vicinity of the FM transition of the Yb moments. The magnitude of the peak increases with increasing value of ΔH for each composition and the position of the maximum shifts from 3 to 9 K when the magnetic field change increases from 1 to 100 kOe. The values of the peak −ΔS^max_M of Yb_1−xMg_xMnO₃ are 3.02 ± 0.37 J mol⁻¹ K⁻¹ for x = 0.00, and 2.63 ± 0.36 J mol⁻¹ K⁻¹ for x = 0.05 with ΔH = 100 kOe. The magnitude of maximum −ΔS_M(−ΔS^max_M) increases with increasing magnetic field.²⁴

Fig. 7 Temperature variation of magnetic entropy change (ΔS_M) for Yb_1−xMg_xMnO₃ (a) x = 0.00 and (b) x = 0.05 samples. Temperature variation of adiabatic temperature change (ΔT_ad) for different field change for Yb_1−xMg_xMnO₃ (c) x = 0.00 and (d) x = 0.05.

The adiabatic temperature rise ΔT_ad is determined from the magnetization and the heat capacity measurements as functions of temperature and magnetic field. The specific heat data (C_P) of Yb_1−xMg_xMnO₃ (x = 0.00 and 0.05) as a function of temperature is already published in ref. 10.

The total entropy S(0, T) under zero magnetic field can be calculated from the heat capacity data from the relation,

ΔT_ad is the isentropic difference between S(0, T) and S(H, T). S(H, T) is obtained by subtracting the corresponding ΔS_mag(H, T) (calculated from the magnetization data above) from S(0, T).^24,25 The temperature dependence of ΔT_ad is plotted in Fig. 7c and d for different fields. The peak of the curve corresponds to ΔT_ad ∼ 8.6 ± 0.95 K for x = 0.00 and ∼9.06 ± 0.96 K for x = 0.05 for a field change of 100 kOe.

The MCE values (magnitudes of |−ΔS_M| and ΔT_ad) determined from magnetization measurements are slightly higher compared to that determined from the heat capacity measurements.¹⁰ This may be due to the difference in measurement conditions;²⁶ magnetization measurements are done under isothermal conditions while heat capacity measurements are done under adiabatic or semi/quasi adiabatic conditions. Also the temperature steps (ΔT) used for magnetization and heat capacity measurements are different. For comparison, we list the magnetocaloric data in Table 1 of various magnetic materials that could be used as magnetic refrigerants.

Table 1 Comparison of different magnetocaloric properties of Yb_1−xMg_xMnO₃ (x = 0.00 and 0.05) samples with reported data for other materials having large MCE values for temperatures below 20 K (unit of ΔS^max_M reported in J kg⁻¹ K⁻¹ is converted to J mol⁻¹ K⁻¹ and unit of RCP reported in J kg⁻¹ is converted to J mol⁻¹ for easy comparison)

Composition/material	TPeak (K)	ΔH (kOe)	ΔS^maxM (J mol^−1 K^−1)	ΔTad (K)	RCP (J mol^−1)	Ref.
YbMnO3	∼7	100	3.02 ± 0.37	8.6 ± 0.95	41 ± 9	Present
Yb0.95Mg0.05MnO3	∼7	100	2.63 ± 0.36	9.06 ± 0.96	40.0 ± 10	Present
Yb0.7Ho0.3MnO3	∼9	100	3.75 ± 0.78		90 ± 27	32
Yb0.8Sc0.2MnO3	∼9	100	1.87 ± 0.31		30.1 ± 8	23
YbMnO3 (single crystal)	∼9	80	∼2.3	∼15	∼26	27
HoMnO3 (single crystal)	∼12	80	∼5.2	∼12.5	∼144	27
DyMnO3 (single crystal)	∼12	80	∼5.5	∼11.5	∼155	27
HoMnO3 (single crystal)	∼9.5	70	∼3.5	∼6.5	∼86	24
EuTiO3	∼10	70	12.1	21	∼124	31
ErMn2Si2	∼6.5	50	8.4	12.9	122	33
TmZn	∼10	70	6.96	11.2	98.9	34
TmAgAl	∼10	70	4.3		96	35
TmZnAl	∼10	70	3.1		76	35
Dy2Cu2O5	∼10	70	6.4		145	36
Ho2Cu2O5	∼14	70	6.7		137	36
ErNiBC	5	50	6.2	8.6	78	37
GdCr2Si2	8	50	4.5		67	38
Er4NiCd	17	50	15.4		500	39

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In addition to the values of ΔS_M and ΔT_ad the relative cooling power RCP is also evaluated to determine the cooling efficiency of these material. RCP is a measure of the quantity of heat transferred by the magnetic refrigerant between hot and cold sinks and is defined as

RCP = |ΔS^max_M| × |δT_FWHM| (4)

The RCP values of Yb_1−xMg_xMnO₃ are 30 ± 6 J mol⁻¹, and 29.0 ± 6 J mol⁻¹ with ΔH = 80 kOe for x = 0.0 and 0.05 respectively. These values are higher than single crystal YbMnO₃ (RCP = 26 J mol⁻¹ with ΔH = 80 kOe).²⁷ The RCP values increase with increasing field for both compounds. For different class of materials, the magnetocaloric data are extensively reviewed by V. Franco et al.²⁸ and Zhong Wei et al.²⁹ for near room temperature range magneto caloric materials and by Ling-Wei Li³⁰ for low temperature range magneto caloric materials. For comparison, we list the magnetocaloric data of present samples in Table 1 together with similar materials which have the potential as magnetic refrigerants for temperatures below 20 K. The MCE parameters |ΔS^max_M|, RCP and ΔT_ad values of the present Yb_1−xMg_xMnO₃ samples are slightly lower compared to several of the intermetallic compounds reported by Ling-Wei Li³⁰ and are significantly lower compared to the EuTiO₃.³¹

Compared to YbMnO₃, the substitution of nonmagnetic Mg ion at the Yb site causes slight reduction of magnetization and hence slight reduction in the values of magnetic entropy change (ΔS^max_M) and RCP. Results of our earlier studies with doping of nonmagnetic Sc and magnetic Ho ions at the Yb site together with the results of present study are compiled in Table 2. Table 2 presents |ΔS^max_M| and RCP values at 100 kOe together with the effective moment (calculated: μ_eff = ((1 − x)μ²_Yb³⁺ + μ²_Mn³⁺)^1/2 for Mg and Sc doping and μ_eff = ((1 − x)μ²_Yb³⁺ + xμ²_Ho³⁺ + μ²_Mn³⁺)^1/2 for Ho doping) and the value of maximum magnetization M (at 2.5 K, 100 kOe) for YbMnO₃ doped with different types (magnetic and non magnetic) of dopants. There is a clear trend between the |ΔS^max_M|, RCP values and the effective moment/maximum magnetization (M) for different series of samples. The |ΔS^max_M|, RCP values are higher for samples having high effective moment (high M).

Table 2 Comparison of |ΔS^max_M| and RCP values of YbMnO₃ doped with different types (magnetic and non magnetic) dopants

Sample	μeff (μB)	M at 2.5 K, 100 kOe (μB/(f.u.))	|ΔS^maxM| at 100 kOe (J mol^−1 K^−1)	RCP at 100 kOe (J mol^−1)
Yb0.70Ho0.30MnO3 (ref. 32)	8.49	3.738	3.75 ± 0.78	90 ± 27
YbMnO3	6.68	1.74	3.02 ± 0.37	41 ± 9
Yb0.95Mg0.05MnO3	6.6	1.698	2.63 ± 0.36	40 ± 10
Yb0.80Sc0.20MnO3 (ref. 23)	6.36	1.274	1.87 ± 0.31	30.1 ± 8

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The magnetic materials with a second order transition generally obey the relation , where h is the reduced field just around T_C , k is a constant, M_s(0) is the saturation magnetization at low T and S(0, 0) is a reference parameter.⁴⁰ Fig. 8a shows the linear dependence of −ΔS^max_M versus h^2/3 which implies the second order transition for Yb_1−xMg_xMnO₃ (x = 0.0 and 0.05).

Fig. 8 (a) Temperature dependence of magnetic entropy change −ΔS^max_M vs. h^2/3. Symbols correspond to data points and the solid lines are linear fit to it. (b) The universal curve behavior of the normalized entropy change curves as a function of the rescaled temperature for different magnetic field for Yb_1−xMg_xMnO₃ (x = 0.0 and 0.05). Symbols correspond to data points and the solid line is the curve representing eqn (6).

For a second order transition, the phenomenological universal curve, suggested by Franco and Conde,⁴¹ converges ΔS_M versus temperature data on a single universal curve if (−ΔS_M/ΔS^max_M) versus a rescaled temperature (θ) is plotted where θ is defined in eqn (5) below,

where T_r1 and T_r2 are temperatures of the two reference points. For the present study, T_r1 and T_r2 are selected such that ΔS_M(T_{r1, 2}) = 1/2ΔS^max_M.⁴² Fig. 8b shows the dependence of −ΔS_M/ΔS^max_M versus θ for typical field changes for Yb_1−xMg_xMnO₃ (x = 0.00 and 0.05). It is clearly seen that the experimental points of the samples distribute on one universal curve that can be well fitted by the Lorentz function:⁴²where the solid line in Fig. 8b represents the curve corresponding to eqn (6).

4. Conclusions

We reported in this work detailed investigations of the magnetic, magnetoelastic and magnetocaloric properties of Yb_1−xMg_xMnO₃ (x = 0.00 and 0.05) using neutron diffraction and magnetization measurements. The lattice parameter a and unit cell volume V decrease continuously with decreasing temperature but close to the Neel temperature T_N1 ≈ 85 K, the Yb ordering temperature T_N2 ≈ 3.5 K for x = 0.00 and T_N1 ≈ 89 K, T_N2 ≈ 3.5 K for x = 0.05 show a magnetoelastic or magnetostriction anomaly. Anomalous variations near the magnetic transition temperatures are observed in the temperature dependence of other structural parameters such as bond lengths, atomic positions etc. From the magnetization measurements, magnetocaloric parameters have been derived. The RCP values of Yb_1−xMg_xMnO₃ are found to be 41 ± 9 J mol⁻¹, and 40.0 ± 10 J mol⁻¹ with ΔH = 100 kOe for x = 0.00 and 0.05, respectively. ΔS^max_M versus h^2/3 shows that these compounds exhibit a second order transition. The universal curve for the normalized entropy change vs. the rescaled temperature for the Yb_1−xMg_xMnO₃ constructed.

Acknowledgements

This work has been supported by UGC-DAE Consortium for Scientific Research, Mumbai Centre, India in the form of a collaborative research scheme (CRS) through project number CRS-M-199. BSB acknowledges UGC-DAE CSR, Mumbai Centre for project fellowship. AKB is thankful to the National Academy of Sciences, India for their support to this work through Senior Scientist Platinum Jubilee Fellowship Scheme.

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