Critical Exponents in Magnetic Transition

The study of critical phenomena in the second order magnetic phase transition, i.e., close to FM T_C in amorphous and crystalline ferromagnets has been the field of interest for a long time to understand the type of magnetic interactions. The famous Landau theory for the second order magnetic phase transition assumed that the Gibb’s free energy is a function of the order parameter and temperature dependent coefficients [120]. The Gibb’s free energy G is written in terms of powers of order parameter. The magnetization M is taken as the order parameter. The expression for G as per Landau theory is given as,

n!>, :# n<6 := Y B:^]Y p:^$Y q ---(1.23)

Here, the coefficients a and b are temperature dependent parameters. The condition for minimum free energy in isothermal system, without any external influence can be written as (∂G/∂M = 0), and it gives rise,

=

: 2B Y 4p:^] ---(1.24)

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According to eq. 1.24, a plot of M² versus H/M should be a straight line with a positive slope [121]. So, for a second order magnetic phase transition, the plot of M² versus H/M is expected to be a straight line, with a positive slope [122]. The spontaneous magnetization, M_s(0, T), the inverse susceptibility, χ₀^-1(T) and the isothermal magnetization at the critical temperature M(H, T=T_C) follow the power law behavior as given below [120],

:r!># s |t|^u t v 0 ---(1.25) _<!># s |t|^w t x 0 ---(1.26) :!=, > ># s =^y t 0 ---(1.27) Here β, γ and δ are critical exponents corresponding to MS, χ0 and isothermal magnetization at TC

and ε = (T−T_C) /T_C is the reduced temperature. According to Arrott and Noakes [123], the mean- field relation (eq.1.24) can be modified for a more general case as given below,

i= :l

/w !> 6 >#

> Y i:

:l^/u ---(1.28) where, T1 and M1 are material dependent constants. For the critical exponent values corresponding to the mean-field model, i.e., β = 0.5 and γ = 1, eq. (1.28) reduces to eq. (1.24).

In the mean-field model (or molecular field model), the magnetic spins interact with one another through a molecular field proportional to the average magnetization. Here, the exchange interaction, J_ij between all the spins S_i and S_j is identical and independent of displacement. This model is appropriate, whenever the interaction is in long range. Chamberlin [124] has proposed the mean-field cluster model to explain the existence of long range interaction in different substances.

For high magnetic anisotropic materials, the critical exponents generally follow the Ising model. This model deals with only one component of the spins. In highly anisotropic materials, a stronger magnetic coupling is seen in one of the components of spin compared to other two components. The γ value corresponding to three dimensional Ising model is 1.25.

Heisenberg model is an isotropic model, where all components of spins play equal role in magnetic interactions. The magnetic interaction energy between spins Si andSj of atoms present at nearest neighboring positions i and j can be written as U = - 2 J Si . Sj, where, J is the exchange integral and is related to the overlap of the charge distributions of the atoms i, j. Such interaction takes place mostly with nearest neighbor magnetic ions and known as short range interaction.

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The list of critical exponents γ, β and δ predicted by different theoretical models are given as follows.

3-D Heisenberg model γ = 1.336, β = 0.365 δ = 4.80 3-D XY model γ = 1.30, β = 0.34 δ = 4.80 3-D Ising model γ = 1.241, β = 0.325 δ = 4.82 Mean-field model γ = 1.00, β = 0.50 δ = 3.0 Tricritical mean-field γ = 1.00, β = 0.25 δ = 5.0

The above critical exponents are related to each other by the Widom scaling relation [125], δ

= 1+ γ/β and follow the static scaling hypothesis.

Fig.1.21: Scaling law for nickel sample using the data of Weiss and Forrer. The left side and right side curves correspond to T < TC and T > TC respectively. ‘t’ denotes the reduced temperature ε = (T – TC)/ TC and M1 is a constant. (Reproduced from Green et al. [12]

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The static scaling hypothesis predicts that M(H, ε) is an universal function of ε and H as given below,

:!=, t#|t|^u z_{P=|t|^!u|w#S --- (1.29)

where, f₊ and f_- are regular analytical functions for ε > 0 and ε < 0, respectively. According to eq. (1.29), plots of M (H, ε)|ε|^-β versus H|ε|^-(β+γ) would lead to universal curves, one for temperatures T > T_C (ε > 0) and the other for T < T_C (ε < 0). A typical scaling hypothesis plot for Ni sample reported by Green et al.[12] from the experimental data of Weiss and Forrer [126]

is shown in Fig.1.21. They used the exponent values of β = 0.4, γ = 1.315 with Curie temperature TC = 353.8 °C.

To exclude the possibility of error in Tc obscuring the critical exponents, the χ_o data could be analyzed independently in terms of Kouvel-Fisher method [127] and according to that, the expression for susceptibility can be written in the following form

_w ^[ --- (1.30) So, the plot of versus T is expected to exhibit a liner behavior with a slope 1/γ and

the intercept in the T axis equal to T_c. Thus T_c and γ can be determined simultaneously.

Critical exponent analysis has been reported in many metallic alloys, metals and spinels such as Cu-Ni, Co-Ni, Fe-Ni, Fe₈₄B₁₆, FeZr, Fe, Co and Ni metal and also in ferrites such as Fe₃O₄, TiFe2O4, MnFe2O4 etc.[128-132]. Critical exponent analysis have carried out by Hiroyoshi et al.

[129] in Ni and other metallic alloys. They have shown the linear behavior of M^2.5 vs (H/M)^0.75 for all the metallic and alloyed samples and it signifies that the type of interactions are same for all the FM alloys. The observed critical exponents were comparable to Heisenberg Model.

Flores et al.[132] studied the phase transition in Ti_0.2Fe_2.8O₄ based ferrite samples by using Arrott plots and Kouvel-Fisher method and, the magnetic interaction was found to be short range Heisenberg type. The reported values of critical exponents on various perovskite colossal magneto-resistance materials have been found to be disparate with values corresponding to both long range and short range FM interactions. Motome et al. [133] estimated the value of β from the Monte-Carlo simulation and predicted that the above class of materials follow three dimensional (3D) Heisenberg model. Lofland et al. [134] and a few other research groups [135, 136] have observed long range interactions in (La, Sr)MnO₃ system, with estimated critical exponent values following the mean-field model. On the other hand, FM transitions in Sr, Ba and

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Ag substituted La-Mn-O compounds have been found to follow either 3D Heisenberg [137-139]

or 3D Ising [140] model. Venkatesh et al. [141] extracted the critical exponents from magnetization, ac susceptibility, resistivity and specific heat measurements on Nd_0.5Sr_0.5MnO₃ single crystalline sample. They reported that, the exponent values were all between mean-field and 3d-Heisenberg models.

Critical behavior of DMS materials such as Ga_1-xMn_xAs was studied theoretically by Priour et al.[142] using the large scale Monte Carlo calculation. They have modeled the magnetic transition in DMS materials in terms of strongly disordered Heisenberg model in 3D lattice. In the presence of strong disorder, there could be an apparent change in critical behavior especially at relatively large ε value. In such case, the expression for χ can be written by taking into account the correction to scaling as follows [142]

χ = χ

o

(ε

^-γ

+ B t

+ C t

Y q )

--- (1.31) Where γ is the genuine critical exponent for χ_o and, y₁ and y₂ are the exponents for correction in the scaling relation. The effective critical exponent, γ_eff can be written as

γ

_~

!t#

^€

γ

^|B‚_|Bƒ^ƒ„…†_„…†^|C‚_|Cƒ„…†^ƒ„…†

--- (1.32)

At ε→0, γeff → γ, the genuine critical exponent of χo.

Fukuma et al.[143] studied the critical exponents behavior in Ge_1-xMn_xTe based DMS materials.

The obtained parameters deduced from Modified Arrott plots and scaling behavior were found to be in good agreement with the theoretically predicted mean field model. However the values of critical exponents corresponding to (Cd,Mn)Sb and (Cd,Mn)Te [144, 145] show the nearest neighbor Heisenberg interaction.

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