Structural Results
Figure 1 portrays the refinement diagram of La0.9†0.1MnO2.9. The obtained diffraction diagrams depict fairly fine lines that describe the good crystallization of materials. The refinement of the crystalline parameters was carried out by the Rietveld method [27]using the FULLPROF program [28]. The phase analysis was conducted in the rhombohedric system with \(\text{R}\stackrel{-}{3}\text{c}\) space group. The value of the accurate fit indicator χ2 justifies the quality of the refinement, whose results are summarized in Table.1.
Crystallites size (Dsc) was computed from the most intense diffraction peak using the Scherrer’s formula[29]:
$${\text{D}}_{\text{S}\text{C}}=\frac{\text{K}}{{\beta }\text{c}\text{o}\text{s}\left({\theta }\right)}$$
3
Where λ is the X-ray radiation wavelength (λCu = 1.5406 Å), K is the scherrer constant equal to 0.9, θ is the diffraction angle for the most intense peak and β is the full width at half maximum (FWHM) of the most intense peak (in radians). ßis indicated by:
$${{\beta }}^{2}= {{\beta }}_{\text{m}}^{2}+ {{\beta }}_{\text{S}}^{2}$$
4
Whereβmis the esperimental full width at half maximum (FWHM) andβSis the FWHM standard silicon sample [30]. The values are oulined in Table.1.
The thermal dependences of magnetization for La0.9†0.1MnO2.9, obtained at H = 0.1 T in both field cooled (FC) and zero field cooled (ZFC) regimes, are reported in Fig. 2(a). This curvereveals the existence of a paramagnetic-ferromagnetic state with the decrease of temperature, which is observed from the Curie temperature determined from the derivative of the M (FC). Figure 2(b) displays the variation of \(\frac{{\text{d}}_{\text{F}\text{C}}}{\text{d}\text{T}}\) intended to determine the value of the Curie temperature, which is equal to 209 K. The inverse of magnetic susceptibility\(\frac{\text{H}}{\text{M}}= {}^{-1} \left(\text{T}\right)\)under 0.1 T for La0.90.1MnO2.9 samples are schematized in Fig. 3(a)for the determination of the value of the Curie constant and Weiss temperature. The inverse of magnetic susceptibility versus temperature abides by the following Curie Weiss Law [31]:
$$= \frac{\text{C}}{\text{T}-{{\theta }}_{\text{C}\text{W}}}$$
5
Where C is the Curie constant and θCW is the Curie Weiss temperature positive, negative or null for the ferromagnetic, antiferromagnetic and paramagntic interactions, respectively. By fitting the linear paramagnetic region of data, C and θCW were obtained. Using the relation of the Curie constant provided by [32], we get:
$$\text{C}= \frac{{\text{N}}_{\text{A}}{{\mu }}_{0}}{3{\text{K}}_{\text{B}}}\left[{\text{g}}^{2 }\text{J}\left(\text{J}+1\right){{\mu }}_{\text{B} }^{2}\right]= \frac{{\text{N}}_{\text{A}}{{\mu }}_{0}}{3{\text{K}}_{\text{B}}}{{\mu }}_{\text{e}\text{f}\text{f}}^{2}$$
6
where:
NA= 6.0221023 mol− 1.
µ0: gyromagnetic ratio, µ0 = 4\({\pi }\)10− 7 H.m.
kB: the Boltzmann consatnt, kB = 1.3806510− 23 J.K− 1.
µB: the Bohr magneton: µB = 9.2710− 27 J.T− 1
where J = L + S with L = 0, an thus\({{{\mu }}_{\text{e}\text{f}\text{f}}=\text{g} {{\mu }}_{\text{B}}\left[\text{S}\left(\text{S}+1\right)\right]}^{1/2}\). The value of \({{\mu }}_{\text{e}\text{f}\text{f}}=4.90 {{\mu }}_{\text{B}}\) for Mn3+and for Mn4+\({{\mu }}_{\text{e}\text{f}\text{f}}=3.87 {{\mu }}_{\text{B}}\). Again, we calculated the theoretical magnetic moment\({{\mu }}_{\text{e}\text{f}\text{f}}^{\text{t}\text{h}\text{e}\text{o}}\)that can be expressed by:
$${{\mu }}_{\text{e}\text{f}\text{f}}^{\text{t}\text{h}\text{e}\text{o}}= \sqrt{\sum {\text{n}}_{\text{i}}{\left({{\mu }}_{\text{e}\text{f}\text{f}}\left(\text{i}\right)\right)}^{2}}$$
7
$${{\mu }}_{\text{e}\text{f}\text{f}}^{\text{t}\text{h}\text{e}\text{o}}= \sqrt{\left(0.7+2{\delta }\right)\left[{{\mu }}_{\text{e}\text{f}\text{f}}^{\text{t}\text{h}\text{e}\text{o}}{\left({\text{M}\text{n}}^{3+}\right)}^{2}\right]+ \left(0.3-2{\delta }\right)\left[{{\mu }}_{\text{e}\text{f}\text{f}}^{\text{t}\text{h}\text{e}\text{o}}{\left({\text{M}\text{n}}^{4+}\right)}^{2}\right]}$$
8
The values of C, θCW,\({{\mu }}_{\text{e}\text{f}\text{f}}^{\text{e}\text{x}\text{p}}\)and\({{\mu }}_{\text{e}\text{f}\text{f}}^{\text{t}\text{h}\text{e}\text{o}}\) for all samples are exhibited in Table.2. The positive values of θCW confirm the presence of FM interaction between spins. The approximate values of\({{\mu }}_{\text{e}\text{f}\text{f}}^{\text{e}\text{x}\text{p}}\) and\({{\mu }}_{\text{e}\text{f}\text{f}}^{\text{t}\text{h}\text{e}\text{o}}\)can be ascribed to the homogeneity of the PM phase.
The insetFig.3 (b)demonstrates the derivative of 1/χ. Departing from this picture we inferred the existence of Griffiths phase, which can be assigned to the appearance of FM spin clusters within the PM region below a characteristic temperature TG = 286 K. The ordering is fully achieved at TC [33].
In order to examine the effect of the magnetic field on the evolution of the magnetic properties, we traced the variation of magnetization as a function of magnetic field under − 2 T and 2 T AT 5 K (Fig. 4).
The first term quantifies the FM contributions while the second represents the AFM contributions (linear contributions. The magnetic parameters extracted from hysteresis loops fitting are sumurized in Table.3.The saturation magnetization Ms and the molecular weight MW (g/mol) are used to calculate the magnetic moments µfu (µB ) according to the following relation.
$${\mu }_{fu}\left({\mu }_{B}\right)=\frac{{M}_{s}(\text{e}\text{m}\text{u}/\text{g})\times MW(\text{g}/\text{m}\text{o}\text{l})}{{\mu }_{B}(\text{e}\text{r}\text{g}/\text{G})\times {N}_{A}\left({\text{m}\text{o}\text{l}}^{-1}\right)}= \frac{{M}_{s}\times MW}{5585}$$
9
Where µB = 0.9274 x 10− 20 erg/G (Bohr magneton) and NA = 6.022 x 1023 mol− 1 (Avogadro’s number).We found that \({\mu }_{fu}\left({\mu }_{B}\right)=3.64 {\mu }_{B}\). Additionaly, we found that the studied material presents a very small coercitive field. Therefore, this material stands for a promising candidate that can be invested in the spintronic and magnetic refregiration applications.
Electronic and optical properties
Figure 5 illustrates the total electronic density states (TDOS), which proves that La0.9†0.1MnO2.9material displays a metallic behavior for the spin up state. Howover, the material behavior, in spin down states, proves to be a semiconductor. The coexistence of these two behaviors proves the semi-metal behavior.As a matter of fact, La0.9†0.1MnO2.9material can serveas an appropriate condidate for electronic applications.
Grounded on the available DOS results, we can estimate the population of the 2p and 3d bands. Thus, we can compute the valence electron numbers through the integration of DOS. The difference between the numbers of spin-up and spin-down leads us to calculate the spin magnetic moment relying on the following equations [34]:
$${N}_{val}\uparrow =\int n\uparrow \left(\in \right)f\left(\in \right)d(\in )$$
10
$${N}_{val}\downarrow =\int n\downarrow \left(\in \right)f\left(\in \right)d(\in )$$
111
Where \(n\uparrow\) and \(n\downarrow\) are the DOS of spins up and down, respectively, \(f\left(\in \right)\) is the Fermi Dirac distribution.
Therefore, the magnetic moment is expressed in terms of :
$$m=g.S.{\mu }_{B}=2*({N}_{val}\uparrow -{N}_{val}\downarrow ){\mu }_{B}$$
12
We found that the magnetization of saturation per molecule formula is 3.3 \({\mu }_{B}\) which very close to that obtained experimentally. Figure 6 presents the electronic band structure in both cases (spin up and spin down). We notice that the studied material has a direct bandgap semiconductor with a 1.7 eV bandgap .
Figure 7 demonstratesthe variation of the absorption coefficient as a function of the photon energy. On the ground of the result depicted in the Bands Structure, and according to the Tauc relation [35], we traced ininset of Fig. 7 the variation of (αhν)1/2 versus the incident light
energy (hν):
$${\left({\alpha }\text{h}{\upvartheta }\right)}^{2}=\text{A}\left(\text{h}{\upvartheta }-\text{E}\text{g}\right)$$
13
Where A is a constant and \(\text{h}{\upvartheta }\) is the photon energy.
Figure 8-a foregrounds the M vs µ0H for different temperatures from 100 K to 282 K with a step 8 K. The saturation was achieved for a magnetic field of 1 T. In order to identify the phase transition nature, a linear fit of M2 vs µ0H /M plot (Fig. 8-b)indicating a positive slope, corrolorates that the transition is a second order according to Banerjee’s criteria [36].
Magnetocaloric Effect
Figure 8-c highlights the magnetic entropy change (-ΔSM) as a function of temperatureand the magnetic field as determined by the Maxwell relation [37]:
$${\varDelta \text{S}}_{\text{M}}\left(\frac{{\text{T}}_{1 }+{\text{T}}_{2 }}{2}\right)=\frac{1}{{\text{T}}_{2 }-{\text{T}}_{1 }}\left[{\int }_{0}^{{{\mu }}_{0}\text{H}}\text{M}({\text{T}}_{2}, {{\mu }}_{0}\text{H}) {{\mu }}_{0}\text{d}\text{H}-{\int }_{0}^{{{\mu }}_{0}\text{H}}\text{M}({\text{T}}_{1}, {{\mu }}_{0}\text{H}) {{\mu }}_{0}\text{d}\text{H} \right]$$
14
The magnetic entropy increased with the increase of magnetic field and had a maximum at T = TC. The entropy is related to the relative cooling power (RCP) by the relation:
$$\text{R}\text{C}\text{P}= \left|-\varDelta {\text{S}}_{\text{M}}^{\text{m}\text{a}\text{x}}\right|\times {\delta }{\text{T}}_{\text{F}\text{W}\text{H}\text{M}}$$
15
WhereδTFWHMis the full width at half maximum of the magnetic entropy change curve. RCP serves to evaluate the magnetic cooling efficiency of a magnetocaloric material. Figure 8-d presents the diagram of \({\delta }{\text{T}}_{\text{F}\text{W}\text{H}\text{M}}\), (\(-\varDelta {\text{S}}_{\text{M}}^{\text{m}\text{a}\text{x}}\)) and RCP as a function of magnetic field. Such similar variation can be considered as a relevant potential candidate material to be used in the cooling system grounded on the magnetic refrigeration[38].
Figure 8-edisplays the variation of (-ΔCP) provided by the relation [39–40]:
$${\varDelta \text{C}}_{\text{P}}\left(\text{T}, {{\mu }}_{0}\text{H}\right)={\text{C}}_{\text{P}}\left(\text{T}, {{\mu }}_{0}\text{H}\right)-{\text{C}}_{\text{P}}\left(\text{T}, 0\right)=\text{T}\frac{\partial (\varDelta {\text{S}}_{\text{M}}\left(\text{T}, {{\mu }}_{0}\text{H}\right)}{\partial \text{T}}$$
16
Departing from this figure, we infer that (-ΔCP) is negative before TC and positive after TC. RCP and (-ΔCP) suffer from the same phenomenon as(\(-\varDelta {\text{S}}_{\text{M}})\), where these two parameters play an intrinsic role in identifying the applicability of the compound in the magnetic refrigeration. The maximum values of\((-\varDelta {\text{S}}_{\text{M}})\),\({\delta }{\text{T}}_{\text{F}\text{W}\text{H}\text{M}}\)and RCP at different magnetic fields are exhibited in Table.4.
Franco et al [41–43] reported a universal behavior of (-ΔSM) for the compounds having a second order transition. The method rests on the collapse of the entropy variation curves after a scaling process, whatever the applied magnetic field. Figure 9 foregrounds the curves of the entropy variation normalized to the fields applied as a function of temperature θ for both samples. θ is determined according to the previous equation:
$${\theta }=\left\{\begin{array}{c}-\frac{\text{T}-{\text{T}}_{\text{C}}}{{\text{T}}_{\text{r}1}-{\text{T}}_{\text{C}}}, T\le {\text{T}}_{\text{C}}\\ \frac{\text{T}-{\text{T}}_{\text{C}}}{{\text{T}}_{\text{r}2}-{\text{T}}_{\text{C}}}, T>{\text{T}}_{\text{C}}\end{array}\right\}$$
17
Tr1 and Tr2 are referred to as reference temperatures, which proves the validity of our data processing for these compounds. Hence, the second order transition is verified.
We illustrate in Fig. 10-a the magnetic isotherm around TC with a step of 5 K in a temperature range from 159 K to 259 K. It was used to more accurately determine the transition temperature and the critical exponentsβ, γ and δ. As a far as this research work is concerned, we investigated the modified Arrott diagram M1/β as afunction of (H/M)1/γ according to the following standard theoritical models, namely Mean field model (β = 0.5 and γ = 1), Tricritical model (β = 0.25 and γ = 1), 3D-Heisenberg model (β = 0.365 and γ = 1.336) and 3D-Ising model ( β = 0.325 and γ = 1.24). Figure 10.b presents the variation of M2as a function of H/M according to the theory of the mean field. The Arrot plots are nonlinear and show upward curvature even at high field. We can deduce that the mean field is not the best model. Figure 10.c-e provides the variation of M(1/β) = f (H/M)(1/γ) according to the other models: namely Tricritical ( β = 0.25 and γ = 1), 3D-Heisenberg ( β = 0.365 and γ = 1.336) and 3D-Ising model ( β = 0.325 and γ = 1.24). In Fig. 10-f, the variation of relative slope RS [RS = slope (T) / slope (TC)] is plotted. For the best model, the RS plot should be linear and close to the horizontal x-axis. The RS plot of the 3D-Ising model approaches horizontal x-axis with RS ~ 1 for each temperature. The RS plots for other models deviate from the main horizontal line.Departing from these initial values of the critical exponents, the first values of MS (T, 0) and\({{\chi }}_{0}^{-1}\left( \text{T}\right)\)of the Arrott-Noakes plot are determined for strong fields.
Figure 11.a displays the fitting of temperature dependent spontaneous magnetization (MS) as well as the inverse magnetic susceptibility (\({{\chi }}_{0}^{-1}\)) graphs obtained from a modified Arrott plot using Eq. 18. The fitting plots with the power laws (Eq. 18) provide the values of β and γ, which are close to those found using the Ising model.
$$\left\{\begin{array}{c}{\text{M} }_{\text{S}}\left(\text{T}\right)= {\text{M}}_{0}{\left(–{\epsilon }\right)}^{{\beta }}, \epsilon <0 forT<{\text{T}}_{\text{C}}\\ {{\chi }}_{0}^{-1}\left(\text{T}\right)= {(\frac{{\text{h}}_{0}}{\text{M}}}_{0}) {\left({\epsilon }\right)}^{{\gamma }}, \epsilon >0 forT>{\text{T}}_{\text{C}}\\ M=D{\text{H}}^{1/{\delta }}, \epsilon =0 for T={\text{T}}_{\text{C}}\end{array}\right\}$$
18
When \({\epsilon }=(\text{T}-{\text{T}}_{\text{C}})/{\text{T}}_{\text{C}}\)
All obtained results are compared to those reported in state of art works and are summarized of table.5.The evolution of the last exponent δ has been traced directly from the M (H, T) isotherm at TC using Eq. 19. The M (H, T) isotherm at TC (Fig. 11.b) is selected as a critical isotherm according to the previous result. The inset of Fig. 11.b represents the fitting of ln (M) versus ln (H).The linear fit is used to estimate the value of δ = 4.13 ± 0.05.Statistically,β, γand δare interrelated according to the Widom scaling relation [44–45]:
$${\delta }=1+ \frac{{\gamma }}{{\beta }}$$
19
Based on the above Windom scaling relation, the values of βand γacquired from the MAP are used to calculate δ.
Next, the M (H, T) isotherms in the vicinity of TC follow the scaling relation, which can be indicated by the scaling hypothesis as:
$$\text{M} \left(\text{H},\text{T}\right)= {\left({\epsilon }\right)}^{{\beta }}{\text{f}}_{\pm }\left(\frac{\text{H}}{{{\epsilon }}^{{\beta }+{\gamma }}}\right)$$
20
Where f+ for T > TC (ε > 0) and f− for T < TC (ε < 0) are fixed rational functions.
In Fig. 11-c, we plotted M/|ε|−β versus H/|ε|− (β+γ) curves with β and γ obtained from the KF method. It is clear that all data fall into one of both parts of the plots. One part for temperatures above TC (T > TC) and the other for temperatures below TC(T < TC). To ensure a good visualization of the separation of curves, we plotted the curves with log–log scale inset (Fig. 11-c). This evidently suggests that the obtained values of the critical exponents and the TC ones, confirm the reliability and the good concordance with the scaling hypothesis. Indeed, all data fall into two distinct branches, one for temperature below TC and the other for temperature above TC.
Thermoelectric properties :
By analyzing the band structures of a material, one can estimate its transport properties using the semi-classical Boltzmann transport theory. The established relations can be implemented to calculate the Seebeck coefficient, electrical conductivity, and thermal conductivity.
$${\sigma }_{\alpha \beta }\left(T,\mu \right)=\frac{1}{\text{e}\text{T}{\Omega }}\int {\sigma }_{\alpha \beta }\left(\epsilon \right)(\epsilon -\mu )\left[-\frac{\partial {f}_{\mu }(T,\epsilon )}{\partial \epsilon }\right]d\epsilon \left(21\right)$$
$${k}_{\alpha \beta }^{0}\left(T,\mu \right)=\frac{1}{\text{e}\text{T}{\Omega }}\int {\sigma }_{\alpha \beta }\left(\epsilon \right){(\epsilon -\mu )}^{2}\left[-\frac{\partial {f}_{\mu }(T,\epsilon )}{\partial \epsilon }\right]d\epsilon \left(22\right)$$
$${S}_{\alpha \beta }\left(T,\mu \right)={\sigma }_{\alpha \beta }{\left(T,\mu \right)}^{-1}{\vartheta }_{\alpha \beta }(T,\mu )$$
23
As depicted in Fig. 12-a, the total electrical conductivity exhibits the typical behavior of a semiconductor. In this scenario, the conductivity remains constant at low temperatures and experiences an exponential increase in the high-temperature range. This is attributed to the fact that free carrier charges possess higher energy to overcome the potential barrier at elevated temperatures. The changes in thermal conductivity as a function of temperature is shown in Fig. 12-b.
The total thermal, electrical conductivity and Seebeck coefficients are given by \(k=k\left(\uparrow \right)+k(\downarrow )\), \(\sigma =\sigma \left(\uparrow \right)+\sigma \left(\downarrow \right)\)and \(S=(S\left(\uparrow \right)\sigma \left(\uparrow \right)+S(\downarrow \left)\sigma \right(\downarrow \left)\right)/[\sigma \left(\uparrow \right)+\sigma \left(\downarrow \right)]\) (Y)] where (\(\uparrow\)) and (\(\downarrow\)) represent the coefficients for spin up and spin down respectively. Utilizing this model, the total Seebeck coefficient (Fig. 12-c) obtained shows an increasing as temperature increase. Figure 13-c illustrates a positive Seebeck coefficient (S) indicating that holes contribution predominantly governs electric conduction [46]. To assess the material's performance in thermoelectric (TE) applications, Fig. 12-d depicts the evolution of the ZT figure of merit as a function of temperature (T). Similar to the Seebeck coefficient, ZT attains its peak at 420 K (ZT=1.1), subsequently decreasing while maintaining a value above 0.75. this obtained results indicating that the material can be considered a viable option for thermoelectric device integration.