Low-Doped Regime Experiments in LaMnO₃ Perovskites by Simultaneous Substitution on Both La and Mn Sites

Abstract

Although extensive substitution studies exist on LaMnO3 perovskites, simultaneous substitution at the La and Mn sites in the low-doping regime is not as common and provides important insights into the subtle balance of various competing effects acting on the crystal structure and the magnetic properties. This chapter presents a study of the evolution of the crystal structure and magnetic properties by simultaneous substitution of magnetic and non-magnetic ions at the perovskite A and B sites, respectively. It will examine some of the ways in which experiments on the evolution of magnetic properties provide a suitable balance scenario between the competitive effects arising from doping at each site. The work discusses the evolution of the Curie-Weiss behavior and the formation of ferromagnetic (FM) clusters above the Curie temperature, whose structure is also dependent on doping.

1. Introduction

Simultaneous substitution in polycrystalline LaMnO₃ on the La³⁺ and Mn³⁺ ions by magnetic Dy³⁺ and non-magnetic Zn ions, respectively, offers an adequate scenario for studies on the evolution of structure and magnetic properties due to competing effects. Low-doped regime in La_{1 − x}Dy_xMn_{1 − y}Zn_yO₃ manganites$—$(La(Dy) Mn(Zn)O₃)$—$means the amount of substitution ($x,y$) is kept in the $0.0–0.1$ regime for both sites. For both $x=0$, $y=0$, the parent compound LaMnO₃ is an insulator composed of ferromagnetic ($a,b$)-planes of Mn³⁺ ions with orbitals configuration {$t_{2g}^3,e_g^1$}, total spin $S=2$, oriented in basal plane, but antiferromagnetically (AF) coupled along the c axis [1]. This antiferromagnetic ground state structure (A-type) due to the Jahn-Teller (J-T) distortion of the Mn³⁺ ions [2, 3, 4] tends to disappear by oxygen excess, and the LaMnO_(3 + δ) manganite becomes ferromagnetic. The magnetic A-type structure occurs because of the different orientations of the $e_g^1$ orbitals within the ($a,b$) plane lead to FM superexchange, while covalent orientation of these orbitals along the c axis leads to AFM superexchange [5, 6, 7], which weakens the ferromagnetism within the ($a,b$)$$ planes, but the AFM coupling along c axis remains unaltered. For LaMnO_(3 + δ), the correlation between crystal structure and magnetism has been discussed in Ref. [8].

For $\mathit{y}\mathit{=}\mathbf{0}$, the magnetic and transport properties of La_{1 − x}Dy_xMnO₃ compounds with nominal stoichiometry $\mathit{x}\mathit{=}\mathbf{0.05}\mathit{,}\mathbf{0.10}$ synthesized by sol-gel method have been studied by the authors in Ref. [9]. Although LaMnO₃ and La_{1 − x}Dy_xMnO₃ compounds were synthesized with the same method, they observed that in La_{1 − x}Dy_xMnO₃ more La-site vacancies appear. This has been attributed to evaporation of La ions by chemical disorder. The La-site deficiencies, as mentioned, increase the FM double exchange (DE) interaction. Simultaneously, the ionic radius difference between La and Dy ions leads to local distortion, in addition to the J-T distortion caused by Mn³⁺ ions reducing the FM interaction between Mn³⁺ and Mn⁴⁺ ions. Furthermore, the large magnetic moment of Dy³⁺ (${\mathit{\mu }}_{\mathbf{eff}}\mathit{\approx }\mathbf{10.83}{\mathit{\mu }}_{\mathit{B}}$) randomly polarizes the magnetic Mn sublattice in favor of ferromagnetism. In slightly Dy-doped LaMnO₃, the paramagnetic Dy ion can be represented as an impurity between the Mn-Mn spin pairs coupling to Mn ions through 3d and 4f electrons. At large Mn-Dy separation, the interaction between 3d and 4f electrons is smaller than (or just comparable) that between nearby Mn-Mn host ions [10, 11, 12, 13], confirmed that through weak coupling between magnetic impurities and their nearest neighbor host ions, the local AFM order in the host spin system will be enhanced. Through small Dy doping, the interactions between Dy impurities are determined by host Mn spins. On the contrary, when the Dy concentration increases, the interaction between the Dy impurities increases, and the lattice becomes much more disordered. Thus, Dy can introduce magnetic disorder that can be important to understand the presence of ferromagnetic clusters above the ferromagnetic ordering.

For $\mathit{x}\mathit{=}\mathbf{0}$, a great deal of work has been conducted concerning the doping of the Mn site in LaMn_{1 − y}Zn_yO₃ manganites with y in the $\mathbf{0.05}–\mathbf{0.4}$ range [14, 15, 16, 17]. Partial substitution of Mn by Zn²⁺ {3d¹⁰} nonmagnetic ions results in the simultaneous presence of Mn³⁺ and Mn⁴⁺ ions, triggering Zener’s DE interaction. Thus, the transformation of Mn³⁺ into Mn⁴⁺ reduces the antiferromagnetism of the A-type structure, inducing 3-dimensional ferromagnetism [15]. Moreover, the Zn dilution of the Mn sublattice locally weakens magnetism and induces disorder. At the same time, the magnetic La(Dy) sublattice interacts with the local field imposed by the ferromagnetic of the Mn sublattice. Depending on the magnetic nature of the La(Dy) sublattice, the interaction can be FM or AFM; in our case, ferromagnetism is obtained.

2. Crystalline structure

X-ray diffraction (XRD) patterns study of the samples of La_{1 −x}Dy_xMn_{1 −y}Zn_yO₃ family (synthesized by solid reaction method) with $x=0.0,0.05,0.1$ and $y=0.0,0.05,0.1$ for samples$x=0,y=0.0$ and $x=0.0$,$y=0.05$ exhibits a trigonal phase, while the other samples correspond to an orthorhombic phase. The effects on the crystal structure through simultaneous substitution on both La and Mn sites are complex. On the one hand, inclusion of Dy [18, 19, 20], with smaller ionic radius than La ions, and inclusion of Zn [16, 17], with a greater ionic radius than Mn ions, introduces distortions in the octahedra and, therefore, in the crystallographic structure. On the other hand, introduction of divalent Zn ions induces the change of Mn³⁺ for Mn⁴⁺ and the J-T distortion of the Mn³⁺ neighboring Zn²⁺ differs from one of the Mn³⁺ no-neighboring Zn²⁺ ions [17]. Figure 1 shows the XRD pattern for$x=0$ due to different Zn substitution: $y=0$ (bottom panel), $y=0.05$ (middle panel), and $y=0.1$ (top panel).

Figure 1.

X-ray diffraction patterns for La_{1 −x}Dy_xMn_{1 −y}Zn_yO₃ (a) $x=0.0,y=0.1$; (b) $x=0.0y=0.05$; and (c) $x=0y=0.0$. Major Miller indices are indicated.

X-ray diffraction data were refined by the Rietvel method—Fullproff program [21]. Information, such as phase structure, identification of planes, cell volumes, and cell densities were obtained. Table 1 presents some crystallographic parameters corresponding to La_{1 −x}Dy_xMn_{1 −y}Zn_yO₃ samples.

Sample	$\mathsf{x}=\mathsf{0}$.0			x = 0.05
	y = 0.0	y = 0.05	y = 0.10	y = 0.0	y = 0.05	y = 0.10
Structure	Trigonal/R$-$3C		Orthorhombic/Pbnm
Space group	167		62
Z	6		4
a ($\mathring{\mathrm{A}}$)	5.5363	5.5412	5.5459	5.5395	5.5354	5.5354
b ($\mathring{\mathrm{A}}$)	5.5363	5.5412	5.5056	5.4965	5.5017	5.5072
c ($\mathring{\mathrm{A}}$)	13.3670	13.3652	7.7995	7.7765	7.7863	7.7923
Cell volume (Å^3)	354.831	355.400	238.1	236.775	237.124	237.542
Density (g/cm^3)	6.785	6.804	6.807	6.865	6.840	6.828

Table 1.

Structural parameters for La_1 − xDy_xMn_1 − yZn_yO₃ samples. Values in white cells are from Ref. [22].

Figure 2 shows a structural phase transition for $x=0.0$ samples, when$y>0.05$ it is from triclinic to orthorhombic. This signals the high structural distortion introduced by Zn ions. Therefore, the sample $x=0.0$, $y=0.1$ relaxes the stress by symmetry breaking. As will be seen later, this behavior is also observed at Dy concentrations for $x=0.05$, 0.10 series, although for these samples, relaxation is achieved by octahedra tilting increases.

Figure 2.

Variation of unit cell parameter with Zn doping ($0.0<y<0.1$). (a) Cell parameter a (circle) and b (triangle); and (b) Cell parameter c (square). $x=0.0$ in all cases.

Use of 3D reconstruction tools of structures (like Visualization for Electronic Structural Analysis) allows obtaining interatomic distances, $\mathrm{Mn}-\mathrm{O}$ and angles, ${\theta }_i$ in the $\mathrm{Mn}-\mathrm{O}-\mathrm{Mn}$ bonds (Figure 3). In the case of manganites, a distortion of structure is directly related to their magnetic properties. This phenomenon, known as the J-T effect, is a structural phase transition driven by the coupling between the orbital state and the vibronic configuration of the crystal lattice. The J-T coupling to the lattice manifests itself in changes in bond lengths $\mathrm{Mn}-\mathrm{O}$ and ${\theta }_i$ angles, as well as in orbital order [3, 4].

Figure 3.

Unit cell of perovskite structure indicating the bond distances, $\left(\mathrm{Mn}/\mathrm{Zn}\right)-\mathrm{O}1$ and $\left(\frac{\mathrm{Mn}}{\mathrm{Zn}}\right)-\mathrm{O}1$ and bond angles, θ₁ and θ₂. Atoms: O (red), Mn/Zn (black), and La/Dy (Green).

Regarding the measurements of structural distortion in perovskites with orthorhombic structure, two relations have been used to evaluate said distortion from the distances of the manganese with their neighboring oxygen$\mathrm{Mn}-\mathrm{O}-\mathrm{Mn}$ in the plane and with the apical oxygen ions. The octahedra distortion, $D_{\mathrm{oct}},$ can be calculated by [23, 24]:

$$\begin{array}{l}{D}_{\mathrm{oct}}=10\times \frac{\left({\sum }_{i=1,\dots 6}\left|\left(\mathrm{Mn}-{\mathrm{O}}_i\right)-{⟨\mathrm{Mn}-\mathrm{O}⟩}_{\text{average}}\right|\right)}{{⟨\mathrm{Mn}-\mathrm{O}⟩}_{\text{average}}},\\ {\mathrm{D}}_{\mathrm{oct}}=1/6\times {\sum }_{i=1,\dots 6}{\left[\frac{\left(\mathrm{Mn}-O_i\right)-{⟨\mathrm{Mn}-\mathrm{O}⟩}_{\text{average}}}{{⟨\mathrm{Mn}-\mathrm{O}⟩}_{\text{average}}}\right]}^2\end{array}$$E1

where$\mathrm{Mn}-{\mathrm{O}}_i$ represents the bond distances between the manganese ion and each of the six oxygens of the octahedra that surround it. These distances will depend, among others, on the ionic radii and mainly on the doping of the ion in site $B$. Figure 2 shows these distances of the manganese $\mathrm{Mn}-{\mathrm{O}}_2$ in the plane and $\mathrm{Mn}-{\mathrm{O}}_1$ with the apical oxygens.

Octahedra distortion, from the energetic point of view known as J-T distortion, is evaluated through the relation [25]:

$${\lambda }_{\mathrm{J}-\mathrm{T}}=\sqrt{\frac{1}{3}\times \left(\sum {\left(\left(\mathrm{Mn}-{\mathrm{O}}_i\right)-{⟨\mathrm{Mn}-\mathrm{O}⟩}_{\text{average}}\right)}^2\right)}$$E2

Although Eqs. (1) and (2) are different ways of evaluating the distortion, both are dimensionless; their range is between [0, 1], and their behavior is similar. In the manganites studied herein, an inverse relationship between distortions and the magnetic moment of individuals or clusters has been observed for$T>\mathit{Tc}$. Where does this structure-magnetism relationship originate? Clearly, the J-T effect results in spontaneous splitting of the energy levels, reducing the total energy of the ${\mathit{Mn}}^{3+}$ also resulting in a spontaneous distortion of the octahedrons that increase their elastic energy at the cost of a reduction in energy of certain electron orbitals, thus, resulting in a net reduction in energy.

The J-T distortions obtained through Eq. (2) showed anomalous behavior (Figure 4): Zn increases in the range of $0<y<0.05$ increases the J-T distortion and for $y>0.05$, a reduction in the J-T distortion is observed (Figure 4a). This is due to the difference between the ionic radii of Zn and Mn ions [16]. However, when $x>0.05$, the accumulated strain of the system is relaxed via octahedral tilting$—$similar to the introduction of dislocation when the maximum stress tolerated is reached in a crystal [22]. Tilting as a relaxation mechanism is observed in Figure 4b, where greater tilting corresponds to samples with lower J-T distortion. The same anomalous behavior is observed by calculating octahedra distortions (Eq. (2)). As will be seen ahead, the activation energy that allows the magnetic exchange is accompanied by a charge exchange facilitated by these lattice distortions.

Figure 4.

(a) Jahn-Teller distortion versus Zn composition for x = 0.10 samples (squares) and x = 0.05 samples (circles); and (b) Jahn-Teller distortion vs. Octahedra tilting, $\omega$, by using $2{\omega }_i+{\theta }_i=180°$ [22].

Local distortions affect directly the magnetic properties as a function of T, that is, exchange constants are proportional to the orthorhombic strain [8]. Also, magnetic moments and activation energies depend on distortion [22].

3. Local magnetism-electron paramagnetic resonance (EPR) analysis

Electron paramagnetic resonance is an important technique to study the microscopic nature of local interactions in magnetic materials and, particularly, in manganites [26], which in many cases show short-range interactions for $T>\mathit{Tc}$. This technique contrasts with the magnetization and susceptibility techniques that provide global information about the exchange mechanisms and the possible spontaneous creation of clusters at high temperatures.

The EPR signal corresponds to ${\mathit{d\chi }}^{″}/\mathit{dH}$, where χ = M/H is the susceptibility. In the paramagnetic region, where $M$ is linear with H, the double integral of the EPR intensity is proportional to the magnetization. From EPR measurements, we can build the dependence of the inverse of the temperature-dependent susceptibility, ${\chi }^{-1}\left(T\right)$.To carry out the EPR measurements, a Bruker ESP-300 spectrometer was used in the radiation X-band with 9.408 GHz frequency and in a temperature range from $10$ to $290\mathrm{K}$. Figure 5a shows the resonant signals for manganite La_0.9Dy_0.1MnO₃ in the temperature range $220<T<290K$. The inset of Figure 5a is the integral of the EPR signal. It is observed that for high temperatures, the intensity of the EPR signal increases in all cases as the temperature decreases. For $T$ > $T_C$, it is found that the intensity of resonance line fits by the expression [28]:

Figure 5.

(a) EPR signal of La_0.9Dy_0.1MnO₃ for different temperatures, Inset. Intensity obtained by integration of the EPR signal; (b) EPR signal intensity as a function of temperature, showing short-range interaction behavior for $T>181\mathrm{K}$ for the sample La_0.9Dy_0.1Mn_0.95Zn_0.05O₃; and (c) resonant field vs. T for LaMn_0.9Zn_0.1O₃ [27].

$$I_{\mathrm{PM}}=I_0{e}^{∆E/k_{B}T}$$E3

where I_PM is the intensity extracted from the resonance line, $I_0$ is a fitting parameter, and $∆E$ is an activation energy.

Figure 5c shows the variation of the resonant field with temperature. A change in the value of $H_r$ is observed for $T>\mathit{Tc}$. This local signal evidences an FM phase in the PM region [27, 29]. This local magnetism is strongly dependent on doping and oxygen content, as Oseroff et al. [28] show in their work on collective spin dynamics above $\mathit{Tc}$ in manganites (La_{1 − x}Ca_xMnO₃).

The peak-to-peak EPR linewidth, $∆H_{\mathit{PP}},$ can also be used to confirm the presence of short-range interacting magnetic entities. In magnetic resonance, the EPR linewidth is related to the relaxation mechanisms of the magnetic units, whether individual spins or spin-coupled systems. A decrease in $∆H_{\mathit{PP}}$ as temperature decreases indicates PM behavior of the sample. However, in the case of La_1xD_xMnZnO manganites, an increase in $∆H_{\mathrm{PP}}$ is observed as temperature decreases, indicating the presence of short-range interactions. In Figure 6a, the arrows indicate the corresponding critical temperatures for each sample. $∆H_{\mathrm{PP}}$ increases as temperature decreases, indicating a greater range of the collective effects, approaching the FM phase. Tovar et al. [8] have found a direct dependence on temperature for the J-T distortion and the EPR linewidth.

Figure 6.

(a) Temperature dependence on the resonance linewidth; the corresponding $T_C$ values of the $x=0.0$, $y=0.05$ (red) and $x=0.05$, $y=0.1$ (black) samples are indicated; and (b) linewidth as a function of temperature or the $x=0.0$, $y=0.05$ sample; red line corresponds to the fitting line to obtain the activation energy value as E_a = 0.068 eV.

The energy transferred in the relaxation process is related to jumps of polarons thermally activated between the Mn⁴⁺ and Mn³⁺ states. Therefore, the jump rate of the charge carriers will determine the half-life of the spin state and, therefore, determines the EPR linewidth and the conductivity. The dependence of the EPR linewidth as a function of temperature in the paramagnetic region can be evaluated by [30]:

$$∆H_{\mathrm{PP}}\left(T\right)=\frac{A}{T}{e}^{-\left(\frac{E_a}{k_{B}T}\right)}$$E4

$E_a$ values showed on Table 2 are obtained by Eq. (4). Previous studies on relaxation modes in mixed-valence manganites have shown that the EPR predominant signal corresponds to ${\mathrm{Mn}}^{3+}-{\mathrm{Mn}}^{4+}$ relaxation and the internal relaxation of the ions through the lattice. Shengelaya et al. [31] showed that the relaxation, R, of the ${\mathrm{Mn}}^{4+}$ (s) with the lattice can be negligible compared with the relaxation ${\mathrm{Mn}}^{3+}$ (σ) with the lattice (L), while the largest signal corresponds to the relaxation ${\mathrm{Mn}}^{3+}-{\mathrm{Mn}}^{4+}$ and ${\mathrm{Mn}}^{4+}-{\mathrm{Mn}}^{3+}$. Thus, a “Bottleneck” is formed, corresponding to the charge transfer between the two magnetic subsystems (3+ and 4+), while a slower relaxation process, ${\mathrm{Mn}}^{3+}$ with the network occurs. A schematic representation proposed is presented in Figure 7.

$\mathrm{Zn}$ content $\left(y\right)$	Dy $x=0.00$	Dy $x=0.05$	Dy $x=0.1$
	Activation Energy $E_a\left(\mathrm{eV}\right)$	Activation Energy $E_a\left(\mathrm{eV}\right)$	Activation Energy $E_a\left(\mathrm{eV}\right)$
0.00	0.0294(8)	0.0656(7)	0.0596(1)
0.05	0.0683(9)	0.0555(9)	0.0584(5)
0.10	0.0487(7)	0.0477(6)	0.0624(6)

Table 2.

Activation energies as a function of Zn doping (y). White cells from Ref. [27].

Figure 7.

A block diagram showing the energy flow paths for the Mn⁴⁺ and Mn³⁺ spin subsystems and the lattice. The relaxation rates $R_{\mathit{\sigma S},}$$R_{\mathit{S\sigma }}$ represent relaxation between the subsystems. The thickness of the arrows is a measure of the magnitude of the particular relaxation rate [31].

Furthermore, the intensity of the EPR signal is proportional to the static magnetic susceptibility, ${\chi }_e$, it is possible to explain the bottleneck regime due to the coupling of the spins (s) and (σ) as:

$${\chi }_{\mathrm{S}}={\chi }_{\mathrm{S}}^o\frac{1+\lambda ´{\chi }_{\mathrm{\sigma }}^o}{1-\lambda {´}^2{\chi }_{\mathrm{\sigma }}^o{\chi }_{\mathrm{S}}^o}$$E5

where ${\chi }_{\mathrm{S}}^o$ and ${\chi }_{\mathrm{\sigma }}^o$ are the ion susceptibilities without exchange of ${\mathrm{Mn}}^{4+}\text{and}{\mathrm{Mn}}^{3+}$, respectively, and λ is the dimensionless coupling constant.

$${\lambda }^{´}=\frac{-\mathit{zJ}}{n{g}_{\mathrm{s}}{g}_{\sigma }{{\mu }_B}^2}$$E6

with $n$ being the number of spins per cm³ and $g_{\mathrm{s}},g_{\sigma }$ being the g-factors of the ${\mathrm{Mn}}^{4+}$ and ${\mathrm{Mn}}^{3+}$ ions, respectively. For $T\gg T_C$, the EPR signal intensity drops faster than predicted by Eq. (5), associated with a transition from the bottleneck to an isothermal regime: $R_{\sigma \mathrm{L}}\gg R_{\sigma \mathrm{s}}$, where the relaxation, $R_{\sigma \mathrm{s}}$, is T independent [31].

For manganites, the contribution of each coexistent phase has been evaluated by means of the EPR technique. In this case, FM clusters also contribute to the magnetization of the material, so that the total magnetization is the result of the PM and FM contributions above $T_C$ by [32]:

$$M\left(H,T\right)=x\left(T\right){\chi }_{\mathrm{PM}}\left(T\right)H+\left[1-x\left(T\right)\right]M_{\mathrm{FM}}\left(H,T\right)$$E7

where $x$ and $\left(1-x\right)$ are the fractions of PM and FM signals, respectively. The intensity of ESR lines are:

$$I_{\mathrm{PM}}\left(T\right)\propto x\left(T\right){\chi }_{\mathrm{PM}}\left(T\right)\text{and}{I}_{\mathrm{FM}}\left(T\right)\propto \left[1-x\left(T\right)\right]M_{\mathrm{FM}}\left(T\right)$$E8

For $T\gg \mathit{Tc}$, the EPR signal consists only of a PM line; then $x\left(T\gg \mathit{Tc}\right)$ = 1, and $M\left(H,T\right)/H={\chi }_{\mathrm{PM}}$, while $x\left(T<\mathit{Tc}\right)$ = 0 and $M\left(H,T\right)/H$ = ${\chi }_{\mathrm{FM}}$.As Eq. (8) shows, it is possible to find the fraction of the FM phase in the samples by subtracting the EPR intensity from the magnetic susceptibility (Figure 8). The inset shows the fraction as a function of temperature.

Figure 8.

(_°) $I_{\mathit{PM}}\left(T\right)$, (solid line) $M/H\left(T\right)$ at 3.5 kG and (•) $I_{\mathit{FM}}\left(T\right)$ vs. $T$ for La_0.75Ca_0.25MnO₃. Adapted from Ref. [32].

Dormann and Jaccarino [33] proposed the following Huber approximation for a coupled system (clusters) in the PM state:

$$∆H_{\mathrm{pp}}\left(T\right)\propto \left[{\chi }_s\left(T\right)/{\chi }_{\mathrm{EPR}}\left(T\right)\right]∆H_{\mathrm{pp}}\left(\infty \right)$$E9

where ${\chi }_s\left(T\right)=C/T$ is the susceptibility of the individual ions ${\mathrm{M}}^{3+}$ and ${\mathrm{M}}^{4+}$; $∆H_{\mathrm{pp}}\left(\infty \right)$ corresponds to spin-only interactions ($∆H_{\mathrm{pp}}\left(\infty \right)=2600\mathrm{G}$ reported by Causa et al. [34] for perovskites with A = La) and ${\chi }_{\mathrm{EPR}}\left(T\right)$ the PM signal of the coupled system.

$$C=N_A{\mu }_{\mathrm{eff}}^2{\mu }_B^2/3k_B$$E10

4. Macroscopic magnetism: $M\left(T,H\right)$

Thereafter, we will refer to one of the most-used macroscopic techniques to characterize magnetic materials: direct measurement of magnetization as a function of temperature and applied field. This technique offers vast information regarding the type of transition [35, 36, 37], transition temperature [38, 39], magnitude of the magnetic moments [22, 38, 40, 41], and the possible presence of clusters in the PM region, which is quite characteristic of manganites. Next, we will present results on the use of this technique in manganites.

4.1 Type of transition

To determine the nature of the FM-PM phase transition (first- or second-order), it is beneficial to use Arrott’s plot [42]: ${\mu }_{0}H/M$ vs. $M^2$. According to the Banerjee criterion curves, showing positive or negative slopes without inflection points are characteristic of second- or first-order transitions, respectively [35, 43]. Figure 9 shows M vs. T and ${\mu }_{0}H/M$ vs. $M^2$ for La_2/3Sr_1/3MnO₃ [43]. In Figure 9b, in Arrot’s plot for ${\mu }_{0}H/M$ vs. $M^2$, the slope is always positive, indicating the second order of the phase transition for this sample.

Figure 9.

(a) Temperature dependence on FC magnetization for La_2/3Sr_1/3MnO₃ under an applied magnetic field of $H=5\mathit{Oe}$; and (b) Arrott’s plot for ${\mu }_{0}H/M$ vs. $M^2$ for La_2/3Sr_1/3MnO₃ [36].

4.2 Critical temperature and magnetic moment

Magnetization vs. T is used to find the Curie temperature, T_c, or Neel temperature, T_N, in manganites; also, this technic provides information about the presence of magnetic clusters in PM phase. Figure 10a presents the curves M vs. T for the LaMn_0.95Zn_0.05O₃ sample, where T_C$=185K$ is obtained from $\mathrm{d}M/\mathrm{d}T$. The effect of doping on the Tc value is generally one of the first factors to be evaluated in manganites. For low Dy doping in La_{1 −x}Dy_xMn_{1 −y}Zn_yO₃, two competing effects have been observed: firstly, the large magnetic moment of Dy favors FM by increasing the value of Tc, and secondly, the small ionic radius of Dy (close to 50% of the La ionic radius) distorts the lattice, destroying the FM of the $\mathrm{Mn}-\mathrm{O}-\mathrm{Mn}$ chains, which leads to a reduced Tc value. As seen in Figure 10b, Zn determines which of the two effects predominates; at low Zn concentrations, the crystallographic distortion by Dy predominates and for concentrations of $\mathrm{Zn}>0.05$, breaking of the $\mathrm{Mn}-\mathrm{O}-\mathrm{Mn}$chains reduces the effect of lattice distortions by Dy and allows the effect due to the magnetic moment of Dy.

Figure 10.

(a) M vs. T curves for $x=0.00,y=0.05$show magnetization measured at field cooling (open) and the zero-field cooling (close), at $H=0.001\mathrm{T}$. Inset. The derivative of magnetization dM/dT vs. T [27]; (b) $T_c$ vs. $\mathrm{Zn}$ composition. Circles for $0.05\mathrm{Dy}$ series and squares for $0.10\mathrm{Dy}$ series [22]; and (c) ${\chi }^{-1}\mathrm{vs}.T$ for $x=0.00,y=0.05$. Linear fit (red) at high temperatures. CW temperatures, ${\theta }_{\mathit{CW}}$, is indicated [27].

In manganites, it has been observed that the inverse of the susceptibility ${\chi }^{-1}$ vs. T does not satisfy the Curie-Weiss law. However, over a wide range of temperatures ($T\gg \mathit{Tc}$), ${\chi }^{-1}\left(T\right)$ is linear and can be described by the Curie-Weiss model (Figure 10c); the Curie constant (Eq. (10)) can be used to estimate the effective magnetic moments, as well as the doping effect on the magnetic moment. Overall, for all the samples, it has been found that the experimental effective moments are higher than the theoretical values [38, 39, 41, 44], evidence of the presence of clusters in the material. For the La_{1 −x}Dy_xMn_{1 −y}Zn_yO₃ manganite, the theoretical m values are expressed by [22]:

$${\mu }_{\mathit{eff}}^{\mathit{th}}=\left(\left(0.86-2y\right){\left[{\mu }_{\mathit{eff}}^{\mathit{th}}\left({\mathit{Mn}}^{3+}\right)\right]}^2+\left(0.12+y\right){\left[{\mu }_{\mathit{eff}}^{\mathit{th}}\left({\mathit{Mn}}^{4+}\right)\right]}^2\right.{\left.+x{\left[{\mu }_{\mathit{eff}}^{\mathit{th}}\left({\mathit{Dy}}^{3+}\right)\right]}^2\right)}^{\frac{1}{2}}$$E11

according to the stoichiometric formula.

To visualize the cluster behavior throughout the nonlinear PM region, we calculated the values of ${\mathrm{\mu }}_{\mathrm{eff}}^{\exp }=2.83\sqrt{\mathrm{C}\left(\mathrm{T}\right)}{\mathrm{\mu }}_{\mathrm{B}}$ for $T>T_C$ from the difference of $∆{\mathrm{\chi }}^{-1}\left(\mathrm{T}\right)/∆\mathrm{T}=1/\mathrm{C}\left(\mathrm{T}\right)$ [38]. Figure 11b shows ${\mathrm{\mu }}_{\mathrm{eff}}^{\exp }$ vs. T for LaMn_{1 −y}Zn_yO₃ that ${\mathrm{\mu }}_{\mathrm{eff}}$ increases when $T$ decreases in the temperature range $\text{corresponding to}\mathrm{a}{\chi }^{-1}\left(\mathrm{T}\right)$ that has a positive curvature. This suggests an increase in the strength of the exchange coupling with T.

Figure 11.

(a) Theoretical and experimental effective magnetic moments at $T\gg T_C$ for LaMn_{1 −y}Zn_yO₃; and (b) experimental effective magnetic moments as a function of temperature on La_0.95Dy_0.05Mn_{1 −y}Zn_yO₃ for $y=0.0$ (black), $y=0.05$ (red), $y=0.01$ (blue) [22] (b).

4.3 Coupled moments in a mean-field approximation

The Mean-field theory of coupled moment pairs in an effective molecular field approximation, $B_e,$ has been discussed in the literature [28, 34, 45]. The molecular field constants, as well as the magnetic susceptibilities that depend highly on the three main crystal axes direction, result in a slight deviation from the Curie Weiss (C-W) law.

The effective field, $B_e=2\left(z-1\right)\mathit{JM}/\left(N{g}^2{{\mu }_B}^2\right)$ on the coupled moment pair corresponds to a molecular field coefficient, $\lambda =2\left(z-1\right)J/N{g}^2{{\mu }_B}^2$. λ differs from the Weiss model in that $z$ is replaced by ($z-1$). In the Constant-coupling approximation (CCA) [34, 46, 47], this local field “aligns” the magnetic moments of some ${\mathrm{Mn}}^{3+}$ and ${\mathrm{Mn}}^{4+}$, resulting in FM clusters ($S_1+S_2$)- where $S_1$ and $S_2$ correspond to the ${\mathrm{Mn}}^{3+}$ and ${\mathrm{Mn}}^{4+}$ spins, respectively, which conform the cluster unit. This can be modeled, even in the paramagnetic region, by a Heisenberg-type isotropic interaction between pairs of ${\mathrm{Mn}}^{3+}$ and ${\mathrm{Mn}}^{4+}$ ions, subjected to the action of the effective field:

$$\mathcal{H}=-2J{S}_1\bullet S_2+g{\mu }_B\left(S_1+S_2\right)\bullet \left[H+B_e\right]$$E12

where $H$ is the external field. Because the chosen pair is arbitrary, they must all have the same magnetic moment as every other pair. This condition requires that [47]:

$$\mathrm{½}\mathrm{N}g{\mu }_B⟨S_z^{´}⟩=M$$E13

with $⟨S_z^{´}⟩=⟨S_1+S_2⟩$. From Eq. (12) [10], the inverse of susceptibility is obtained as:

$${\chi }^{-1}=\left(T-\frac{1}{2}\mathit{zJ}/k_B\right)/C$$E14

Expanding Eq. (14) around $\left(j-j_C\right)$, where $j=J/k_{B}T$ is the reduced exchange constant, we obtain an expression for the inverse of the susceptibility in terms of $T_c$:

$${\chi }^{-1}=\left(T-T_c\right)/\mathit{\rho C}$$E15

with $\rho =\frac{2}{\left[J_c(2\left(z-1\right)J_c+\left(z-4\right)\right]}$. This expression reproduces a deviation from the linear C-W behavior for FM interactions.

A Heisenberg model over all points of the magnetic lattice $\left(i,j\right)$ is insoluble because said magnetic solid has on the order of ${10}^{22}$ magnetic moments,

$$H=-\sum _{\mathit{ij}}{J}_{\mathit{ij}}{S}_i\bullet S_j$$E16

If we consider only first neighbors in the FM Heisenberg Hamiltonian, it can be rewritten as [48]:

$$\mathcal{H}=-2J\sum _{n,n´}{S}_n\bullet S_{n´}-g{\mu }_{B}H\sum _n{S}_n^z$$E17

From a power series expansion in $j=J/k_{B}T$ [49], the susceptibility is obtained as:

$$\chi =\frac{N{g}^2{{\mu }_B}^{2}S\left(S+1\right)}{3k_{B}T}\sum _{l=0}^{\infty }{a}_l{j}^l$$E18

with $\mathit{N}$ being the number of sites in the sample, $\mathit{g}{\mathit{\mu }}_{\mathit{B}}\mathit{S}$ the magnetic moment associated with the spin, $\mathit{S}$ at each lattice point, $\mathit{z}$ the coordination number, and j being a dimensionless value: $\mathit{j}\mathit{>}\mathit{0}$ for FM and $\mathit{j}\mathit{<}\mathit{0}$ for AFM. Finally, Eq. (18) is rewritten as [48]:

$${\chi }_0^{-1}=\frac{\mathrm{T}}{\mathrm{C}}{\left(1-\mathrm{\tau }\right)}^{4/3}\mathrm{f}\left(\mathrm{\tau }\right)$$E19

with $\tau =T_C/T$, $f\left(\tau \right)=\left\{\left(1-b_1\tau \right)\left(1-b_2\tau \right)/\left(1-a_1\tau \right)\left(1-a_2\tau \right)\left(1-a_3\tau \right)\right\}$ and $a_i,b_i$ parameters obtained by the Padé approximants that only depend on the crystal structure and on the S value. $a_i,b_i$ parameters for $z=6$ are presented in Table 3.

S	$a_1$	$a_2$	$a_3$	$b_1$	$b_2$
3/2	−1/21,9677	−1/1.62282	−1/0.38910	−1/1.74300	1/0.38911
2	−1/51.0423	−1/1.73724	*	−1/1.89133	*

Table 3.

$a_i,b_i$ coefficients obtained for Z = 6, from Padé approximants.

Corresponding to conjugate values reported by Gammel et al. [48] (* $a_3=b_2)$.

With the coefficients indicated in Table 3, Eq. (19) shows a deviation from the straight lines predicted from the C-W theory (Figure 12a). The blue line represents experimental data for La_0.9Dy_0.1MnO₃, where clusters are present at $T>T_C$ and $J$ is unknown. For general Heisenberg Hamiltonians, where more than one relevant exchange constant $J_i(J_i$ =1, 2, 3, 4) is required, a high-temperature expansion (HTE) has been developed [50, 51]. Figure 12b shows ${\chi }^{-1}\mathrm{vs}.T$ experimental and fitting data for La_0.95Dy_0.05Mn_0.9Zn_0.1O₃. The J value obtained is $J/K_B=15K$. The effect of Dy and Zn doping is evident on $J$ (see Figure 12c) [27].

Figure 12.

${\chi }^{-1}\mathit{vs}.T$ (a) using Eq. (19) for $z=\mathbf{6}$, $\mathit{s}=\mathbf{3}/\mathbf{2}$ (black line), $\mathit{S}=\mathbf{2}$ (red line) and the experimental values [22] for ${\mathit{La}}_{\mathbf{0.9}}{\mathit{Dy}}_{\mathbf{0.1}}\mathit{Mn}{\mathit{O}}_{\mathbf{3}}$ (blue line); (b) La_0.95Dy_0.05Mn_0.9Zn_0.1O₃ [22]. Black line, experimental data; red line, fit to HTE algorithm; dashed line, C-W model. $\mathit{J}$ exchange constant values for both series from HTE algorithms; and (c) $\mathit{J}$ values vs. y for La_{1 −x}Dy_xMn_{1 −y}Zn_yO₃ [27].

Acknowledgments

The authors thank the BC foundation for its support in measuring the magnetic properties of the manganites and thank José Fernando López Toro for authorizing the use of diverse results from his Ph.D. thesis to illustrate the analysis presented herein. This publication was partially supported by the Science Faculty and physics department of the Universidad Nacional de Colombia.

Conflict of interest

The authors declare no conflict of interest.