**4. Macroscopic magnetism:** *M T*ð Þ **,** *H*

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 [35–37], 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]: *μ*0*H=M* vs. *M*<sup>2</sup> . 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 *μ*0*H=M* vs. *M*<sup>2</sup> for La2/3Sr1/3MnO3 [43]. In **Figure 9b**, in Arrot's plot for *μ*0*H=M* vs. *M*<sup>2</sup> , the slope is always positive, indicating the second order of the phase transition for this sample.

#### **4.2 Critical temperature and magnetic moment**

Magnetization vs. *T* is used to find the Curie temperature,*Tc*, or Neel temperature, *TN*, 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 LaMn0.95Zn0.05O3 sample, where *TC*¼ 185 *K* is obtained from d*M=*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 La1 � *<sup>x</sup>*Dy*x*Mn1 � *<sup>y</sup>*Zn*y*O3, 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 Mn � O � 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 Zn >0*:*05, breaking of the Mn � O � Mn chains reduces the effect of lattice distortions by Dy and allows the effect due to the magnetic moment of Dy.

#### **Figure 9.**

*(a) Temperature dependence on FC magnetization for La2/3Sr1/3MnO3 under an applied magnetic field of <sup>H</sup>* <sup>¼</sup> <sup>5</sup> *Oe; and (b) Arrott's plot for <sup>μ</sup>*0*H=M vs. M*<sup>2</sup> *for La2/3Sr1/3MnO3 [36].*

*Low-Doped Regime Experiments in LaMnO3 Perovskites by Simultaneous… DOI: http://dx.doi.org/10.5772/intechopen.107309*

#### **Figure 10.**

*(a)* M *vs.* T *curves for x* ¼ 0*:*00,*y* ¼ 0*:*05 *show magnetization measured at field cooling (open) and the zero-field cooling (close), at H* ¼ 0*:*001 T*. Inset. The derivative of magnetization d*M*/d*T *vs.* T *[27]; (b) Tc vs.* Zn *composition. Circles for* 0*:*05 Dy *series and squares for* 0*:*10 Dy *series [22]; and (c) χ*�1vs*:T for x* ¼ 0*:*00,*y* ¼ 0*:*05*. Linear fit (red) at high temperatures. CW temperatures, θCW , is indicated [27].*

In manganites, it has been observed that the inverse of the susceptibility *χ*�<sup>1</sup> vs. T does not satisfy the Curie-Weiss law. However, over a wide range of temperatures (*<sup>T</sup>* <sup>≫</sup> *Tc*), *<sup>χ</sup>*�1ð Þ *<sup>T</sup>* 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 La1 � *<sup>x</sup>*Dy*x*Mn1 � *<sup>y</sup>*Zn*y*O3 manganite, the theoretical m values are expressed by [22]:

$$\mu\_{\rm eff}^{\rm h} = \left( (\mathbf{0.86} - 2\mathbf{y}) \left[ \mu\_{\rm eff}^{\rm h} (\mathbf{M} n^{3+}) \right]^2 + (\mathbf{0.12} + \mathbf{y}) \left[ \mu\_{\rm eff}^{\rm h} (\mathbf{M} n^{4+}) \right]^2 + \propto \left[ \mu\_{\rm eff}^{\rm h} (\mathbf{D} \mathbf{y}^{3+}) \right]^2 \tag{11}$$

according to the stoichiometric formula.

To visualize the cluster behavior throughout the nonlinear PM region, we calculated the values of μexp eff <sup>¼</sup> <sup>2</sup>*:*<sup>83</sup> ffiffiffiffiffiffiffiffiffiffi C Tð Þ <sup>p</sup> <sup>μ</sup><sup>B</sup> for *<sup>T</sup>* <sup>&</sup>gt; *TC* from the difference of <sup>Δ</sup>χ�<sup>1</sup>ð Þ <sup>T</sup> *<sup>=</sup>*Δ<sup>T</sup> <sup>¼</sup> <sup>1</sup>*=*C Tð Þ [38]. **Figure 11b** shows <sup>μ</sup>exp eff vs. *T* for LaMn1 � *<sup>y</sup>*Zn*y*O3 that <sup>μ</sup>eff increases when *<sup>T</sup>* decreases in the temperature range corresponding to a *<sup>χ</sup>*�<sup>1</sup>ð Þ <sup>T</sup> that has a positive curvature. This suggests an increase in the strength of the exchange coupling with *T*.

#### **4.3 Coupled moments in a mean-field approximation**

The Mean-field theory of coupled moment pairs in an effective molecular field approximation, *Be*, 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, *Be* <sup>¼</sup> <sup>2</sup>ð Þ *<sup>z</sup>* � <sup>1</sup> *JM<sup>=</sup> Ng*<sup>2</sup>*μ<sup>B</sup>* <sup>2</sup> ð Þ on the coupled moment pair corresponds to a molecular field coefficient, *<sup>λ</sup>* <sup>¼</sup> <sup>2</sup>ð Þ *<sup>z</sup>* � <sup>1</sup> *<sup>J</sup>=Ng*<sup>2</sup>*μ<sup>B</sup>* <sup>2</sup> . 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 Mn<sup>3</sup><sup>þ</sup> and Mn<sup>4</sup>þ, resulting in FM clusters (*S*<sup>1</sup> <sup>þ</sup> *<sup>S</sup>*2)- where *<sup>S</sup>*<sup>1</sup> and *<sup>S</sup>*<sup>2</sup> correspond to the Mn<sup>3</sup><sup>þ</sup> and Mn<sup>4</sup><sup>þ</sup> 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 Mn<sup>3</sup><sup>þ</sup> and Mn<sup>4</sup><sup>þ</sup> ions, subjected to the action of the effective field:

$$\mathcal{H} = -\mathcal{Q}\mathbf{S}\_1 \bullet \mathbf{S}\_2 + \mathbf{g}\mu\_B(\mathbf{S}\_1 + \mathbf{S}\_2) \bullet [H + B\_\epsilon] \tag{12}$$

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]:

$$\text{Wg}\,\text{Ng}\,\mu\_B\left<\text{S}\_x\right> = M \tag{13}$$

with *S*´ *z* � � <sup>¼</sup> h i *<sup>S</sup>*<sup>1</sup> <sup>þ</sup> *<sup>S</sup>*<sup>2</sup> . From Eq. (12) [10], the inverse of susceptibility is obtained as:

*Low-Doped Regime Experiments in LaMnO3 Perovskites by Simultaneous… DOI: http://dx.doi.org/10.5772/intechopen.107309*

#### **Figure 11.**

*(a) Theoretical and experimental effective magnetic moments at T* ≫ *TC for LaMn1* � <sup>y</sup>*Zn*y*O3; and (b) experimental effective magnetic moments as a function of temperature on La0.95Dy0.05Mn1* � <sup>y</sup>*Zn*y*O3 for y* ¼ 0*:*0 *(black), y* ¼ 0*:*05 *(red), y* ¼ 0*:*01 *(blue) [22] (b).*

$$\chi^{-1} = \left(T - \frac{1}{2}\mathbf{z}\mathbf{J}/k\_B\right)/\mathbf{C} \tag{14}$$

Expanding Eq. (14) around *j* � *j C* , where *<sup>j</sup>* <sup>¼</sup> *<sup>J</sup>=kBT* is the reduced exchange constant, we obtain an expression for the inverse of the susceptibility in terms of *Tc*:

$$\chi^{-1} = (T - T\_c) / \rho \mathbf{C} \tag{15}$$

with *<sup>ρ</sup>* <sup>¼</sup> <sup>2</sup> *Jc* <sup>2</sup>ð Þ *<sup>z</sup>*�<sup>1</sup> *<sup>J</sup>* ½ � <sup>ð</sup> *<sup>c</sup>*þð Þ *<sup>z</sup>*�<sup>4</sup> . This expression reproduces a deviation from the linear C-W behavior for FM interactions.

A Heisenberg model over all points of the magnetic lattice ð Þ *i*, *j* is insoluble because said magnetic solid has on the order of 10<sup>22</sup> magnetic moments,

$$H = -\sum\_{\vec{\eta}} J\_{\vec{\eta}} \mathbf{S}\_i \bullet \mathbf{S}\_j \tag{16}$$

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

$$\mathcal{H} = -2J\sum\_{n,n'} \mathbf{S}\_n \bullet \mathbf{S}\_{n'} - \mathbf{g}\mu\_B H \sum\_n \mathbf{S}\_n^{\mathbf{e}} \tag{17}$$

From a power series expansion in *j* ¼ *J=kBT* [49], the susceptibility is obtained as:

$$\chi = \frac{\text{Ng}^2 \mu\_B \text{\*} \text{S} (\text{S} + \text{1})}{\text{3k}\_B T} \sum\_{l=0}^{\text{es}} a y^l \tag{18}$$

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

$$\chi\_0^{-1} = \frac{\mathbf{T}}{\mathbf{C}} (\mathbf{1} - \mathbf{\tau})^{4/3} \mathbf{f}(\mathbf{\tau}) \tag{19}$$

with *τ* ¼ *TC=T*, *f*ð Þ¼ *τ* f g ð Þ 1 � *b*1*τ* ð Þ 1 � *b*2*τ =*ð Þ 1 � *a*1*τ* ð Þ 1 � *a*2*τ* ð Þ 1 � *a*3*τ* and *ai*,*bi* parameters obtained by the Padé approximants that only depend on the crystal structure and on the S value. *ai*,*bi* parameters for *z* ¼ 6 are presented in **Table 3**.

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 La0.9Dy0.1MnO3, where clusters are present at *T* > *TC* and *J* is unknown. For general Heisenberg Hamiltonians, where more than one relevant exchange constant *Ji Ji* ð =1, 2, 3, 4) is required, a high-temperature expansion (HTE) has been developed [50, 51]. **Figure 12b** shows *χ*�1vs*:T* experimental and fitting data for La0.95Dy0.05Mn0.9Zn0.1O3. The *J* value obtained is *J=KB* ¼ 15 *K*. The effect of Dy and Zn doping is evident on *J* (see **Figure 12c**) [27].


**Table 3.** *ai*,*bi coefficients obtained for* Z *= 6, from Padé approximants.*

*Low-Doped Regime Experiments in LaMnO3 Perovskites by Simultaneous… DOI: http://dx.doi.org/10.5772/intechopen.107309*

**Figure 12.**

*<sup>χ</sup>*�<sup>1</sup> *vs:T (a) using Eq. (19) for z* <sup>¼</sup> **<sup>6</sup>***, <sup>s</sup>* <sup>¼</sup> **<sup>3</sup>***=***<sup>2</sup>** *(black line), <sup>S</sup>* <sup>¼</sup> **<sup>2</sup>** *(red line) and the experimental values [22] for La***<sup>0</sup>***:***<sup>9</sup>***Dy***<sup>0</sup>***:***<sup>1</sup>***MnO***<sup>3</sup>** *(blue line); (b) La0.95Dy0.05Mn0.9Zn0.1O3 [22]. Black line, experimental data*; r*ed line, fit to HTE algorithm*; d*ashed line, C-W model. J exchange constant values for both series from HTE algorithms; and (c) J values vs.* y *for La1* � <sup>x</sup>*Dy*x*Mn1* � <sup>y</sup>*Zn*y*O3 [27].*
