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Critical Behavior at Surfaces of Strong Coupling Paramagnetic Systems Exhibiting a Paramagnetic-Ferrimagnetic Transition

A. Waddahou, M. Chahid*, A. Maârouf, F. Benzouine

Polymer Physics and Critical Phenomena Laboratory, Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, P. O. Box 7955, Casablanca, Morocco

Abstract: The purpose of this work is the investigation of the critical surface effects of two strongly coupled paramagnetic sublattices exhibiting a para-ferrimagnetic transition. The model is of Landau-Ginzburg type, whose bulk free energy is a functional of two kind of order parameters (local magnetizations) φ and This free energy involves, beside quadratic and quartic terms in both φ and , a lowest-order coupling, where Co<0 is the coupling constant measuring the interaction between the two sublattices. Two terms Hφ and are also introduced, to describe the interaction within an external magnetic field H. We introduce a surface free energy expanded in terms of the local order parameters. The magnetization at the surface are φs and We show, in particular, that the model can be reduced to an effective theory written in terms of the overall magnetization and the associated fraction of magnetization . This formulation leads us to define an effective extrapolation length λs. We then derive all the critical properties of the system close to the critical temperature Tc. In particular, we determine the critical behavior of the overall surface magnetization , in terms of b above and below Tc. The variations of s with the magnetic field H, and when a surface field Hs is applied actually at Tc, are derived. We determine, also, the associated susceptibilities at the surface χs and χs.s. The determination of the full profile of the magnetization close to the surface will be the subject of a future communication.

Keywords: sublattices, coupling, paramagetism, ferrimagnetism, transition, critical surface effects

Citation: A. Waddahou, M. Chahid, A. Maârouf, F. Benzouine, Critical Behavior at Surfaces of Strong Coupling Paramagnetic Systems Exhibiting a Paramagnetic-Ferrimagnetic Transition. International Journal of Physics, 6 (4). 116-121. January 2018. doi:10.12691/ijp-6-4-3

1. Introduction

Magnetic and structural properties of the so-called super-weak ferrimagnetic systems are a subject of a great deal of attention from theoretical and experimental point of view. This is due to their considerable importance, especially, in the domain of energy stocking (long life lithium batteries). They may exhibit a para-ferrimagnetic transition at a critical temperature greater than room temperature. Among these, we can quote certain members of Heusler Pauli-paramagnetic alloys [1] based on the composition , with and lamellar Curie-Weiss paramagnetic compounds [2], like with . The common feature of these materials is that they present a small magnetization at low temperature, in contrary to the usual ferrimagnetic materials.

To study the critical magnetic behavior of such materials, a continuous model based on the landau theory [3,4,5] has been successfully used by Neumann and co-workers [6]. Such a model assumes that the material consists of a lattices made up of two coupled Pauli or Curie-Weiss paramagnets sublattices [7,8], with respective local magnetizations and Above the critical temperature both magnetizations vanish and the system is a paramagnet. Below this temperature an antiparallel configuration of the magnetization is favored, but with non-vanishing overall magnetization. One can say that the material exhibits a ferrimagnetic state. Quantitatively, this coupling manifests itself through the introduction of an extra term in the free energy. Negative values of the coupling constant favor the anti-parallel alignment of the local moments and and ferrimagnetic order appears. Within the framework of this model the par-ferrimagnetic transition in the bulk arising from these materials was widely studied applying, first, from a mean-field point of view, the theory has been developed through numerical method [6], and through an exact analytic analysis [9,10,11]. Second, using the Renormalization-Group techniques [12,13] as in the case of usual para-ferromagnetic transition [14,15,16].

The purpose of this work is to investigate the critical properties of the system at surfaces. Indeed, near a second order phase transition point, the correlation length or order parameters fluctuations becomes long-ranged, and hence the effect of surfaces on the bulk properties of the system is much more drastic. We will focus our attention to the situation where the surface still makes a small contribution to any bulk property only. Then it makes sense to split the free energy, in a bulk term proportional to the volume and a surface term proportional to the surface area. We first, reformulate the model in terms of an effective -theory using the fraction of the magnetization , and the order parameter of interest is then the overall magnetization . Under these considerations the effective phenomenological constants of the free energy involve information about competition between the coupling and temperature. The surface free energy is expanded in terms of the fraction of the magnetization at surface with . This leads us to define an effective extrapolation length (see below). We determine, in particular, the overall magnetization at surface as function of , the bulk correlation length , and the overall bulk magnetization . In addition to these quantities, depends, also, on the surface and bulk square masses and , that reflect the dependence on both temperature and coupling . We determine, also, the local susceptibilities at the surface. The first is the response of a surface spin to a uniform field acting throughout the system, and the second is the response to a field acting in a surface.

The remainder of presentation proceeds as follows. Section to is devoted to a description of the used model. The reformulation of this latter as an effective -theory, and the investigation of the mean-field critical properties at surface of the system is the aim of Section 3. We draw some concluding remarks in section 4.

2. The Model

The physical system we consider here consists of two strongly coupled sublattices, of respective moments and . For small moments and in the presence of an applied external magnetic field , in a Landau approximation, the bulk free energy allowing to investigate the para-ferrimagnetic transition within this system writes [6,9,10]

[picture] | (2.1)

The squared gradient terms on the right-hand side of relation (2.1) traduce the spatial variations of order parameters and . There, stands for the d-dimensional position vector of the considered point. In relation (2.1), the coupling constants and are taken to be positive, to ensure the stability of the free energy. Coefficients and depend on temperature according to:

[picture] | (2.2A)

for a Pauli paramagnet [7,8], or

[picture] | (2.2b)

for a Curie-Weiss paramagnet [8,17]. and appearing in relation , have a simple dependence in both free electron density and Fermi energy relative to the two sublattices [18]. In relation the Curie-Weiss temperatures and are proportional to exchange integrals and , inside the sublattices [12]. The extra term in Eq. (2.1) represent the lowest-order coupling between the two sublattices. Such a term plays, in fact, the role of an internal magnetic field. For a negative coupling constant an antiparallel configuration of magnetizations and is favored, while favors their parallel alignment. For Curie-Weiss materials like lamellar compounds [2,19], the coupling is proportional to the exchange integral between the two sublattices. In this work we are concerned only with negative values of in order to investigate the ferrimagnetic state of the system. is a (suitable normalized) magnetic field. For a film of thickness the generalization of Eq. is

[picture] | (2.3)

where similarly the surface free energy is expanded in terms of the local order parameters including terms up to second order-only

[picture] | (2.4)

We wish to study the system close enough to we then neglect higher terms in in Eq. . The linear term involves a field acting on spins in the surface plane only, and the constants of the quadratic terms were written arbitrarily as where the parameters have the dimension of a length and are called extrapolation lengths [20]. Next we consider the case where and are homogeneous and disregard variations of the magnetization within the layers, hence replacing and by their averages over , which we denote simply by and . Equations and then become

[picture] | (2.5)

Functional differentiation of Eq. yields ( being the total surface area of the film)

[picture] | (2.6a)

[picture] | (2.6b)

for which the surface terms in Eq. supply the boundary conditions

[picture] | (2.7a)

[picture] | (2.7b)

and

[picture] | (2.8a)

[picture] | (2.8b)

It is important to recall that some previous results [9] can be obtained from Eqs. , for the overall bulk magnetization and the overall bulk susceptibility of a homogeneous system where and can be omitted. The first important result is that the critical temperature occurs at [9]

[picture] | (2.9)

Just below the critical temperature , the overall magnetization is given by

[picture] | (2.10)

and the overall susceptibility is given by (superscripts + and – distinguish temperatures above and below )

[picture] | (2.11a)

[picture] | (2.11b)

We recall also that the correlation length is given by [9]

[picture] | (2.12)

3. Effective Theory and Critical Behavior at Surface

We focus, for the moment, our attention to a semi-infinite system, ; in this case the phase transition in the bulk of the film occurs at precisely the critical temperature as for a fully infinite system, and we may replace the second boundary condition, Eqs. , by

[picture] | (3.1)

We can obtain the magnetizations at the surface and multiplying respectively Eqs. and , by and respectively, and integrating over from zero to infinity, whereby the boundary conditions Eqs. and , can be used. This leads to the following two coupled equations

[picture] | (3.2a)

[picture] | (3.2b)

Making the sum of Eqs and , we have then

[picture] | (3.3)

[picture] | (3.4a)

[picture] | (3.4b)

where and can be viewed as the fractions of bulk and surface magnetizations relatively to the two sublattices. With help of these changes and using relation , Eq. becomes

[picture] | (3.5)

where is an effective constant and the effective extrapolation length, which write

[picture] | (3.6a)

and

[picture] | (3.6b)

It is important to note that we can write the model as an effective -theory in terms of the fraction of magnetizations and the overall magnetization Indeed, under these considerations the free energy reduces to

[picture] | (3.7)

with

[picture] | (3.8a)

[picture] | (3.8b)

[picture] | (3.8c)

and

[picture] | (3.9)

where and are those parameters given by Eqs. . Functional differentiation of Eq. yields an equation due to Ginsburg Landau [21] and familiar from theory of superconductivity, but with the effective phenomenological parameters , and . Notice that these latters contain all information about the system (like dependence in temperature, coupling…).

[picture] | (3.10)

for which the surface term in Eq. supply the boundary conditions

[picture] | (3.11a)

[picture] | (3.11b)

From Eq. one obtains the standard results for the bulk magnetization and the overall susceptibility , for a homogeneous system where can be omitted

[picture] | (3.12)

with

[picture] | (3.13a)

[picture] | (3.13b)

The fraction of bulk magnetization is that given by relation . Note that close to the critical temperature , the effective phenomenological parameter behaves as: . Indeed at , we have: , which can be written as de product of the two following equations

[picture] | (3.14a)

[picture] | (3.14b)

Reporting these expressions of in relation , one obtains The overall susceptibility above and below , can be obtained directly by taking the first derivative of Eq. with respect of the magnetic field , and setting , we find

[picture] | (3.15a)

[picture] | (3.15b)

Expressions of the correlation length above and below can also be extracted

[picture] | (3.16a)

[picture] | (3.16b)

where . Remark that the behavior of these bulk quantities Eqs. and , obtained from the effective theory are equivalent to those derived above from the theory written as two coupled order parameters and given by Eqs. . It is also important to note that for a semi-infinite system, , we may replace the boundary condition Eq. , by

[picture] | (3.17)

and the magnetization at the surface is given by: Multiplying Eq. by and integrating over from zero to infinity, whereby the boundary conditions Eqs. and , can be used. We obtain directly the equation which writes

[picture] | (3.18)

To solve this latter we first assume positive surface energy, . Then a spontaneous magnetization at the surface exists only for , where a spontaneous magnetization also exists in the bulk. Equation yields

[picture] | (3.19)

where and are those surface parameters given by

[picture] | (3.20a)

[picture] | (3.20b)

The solution of Eq. writes then

[picture] | (3.21)

which for reduces to

[picture] | (3.22)

where and are given by

[picture] | (3.23a)

[picture] | (3.23b)

Expression shows that the magnetization at surface vanishes linearly at the critical temperature . Notice that for a vanishing coupling constant where the two sublattices are decoupled, the ratio reduces to which is the result relative to the usual [20]. It is also straightforward to obtain the variation of the bulk magnetization with the magnetic field actually at (where ). Indeed at the temperature parameter , from Eq. one obtain

[picture] | (3.24a)

and using Eq. , writes then

[picture] | (3.24b)

In contrast, if a surface field is applied only exactly at , it cannot induce any magnetization in the bulk and the magnetization at the surface is simply given by

[picture] | (3.25)

Next we obtain the susceptibilities at the surface, considering the linear response of with respect to both and , above and below . Inserting in Eq. (3.18), and using Eq. (3.15a) for and , yields

[picture] | (3.26a)

[picture] | (3.26b)

and using for and close to , yields

[picture] | (3.27a)

[picture] | (3.27b)

Similarly, one can obtain the response to the local field (at ). Above the critical temperature and from Eq. surface magnetization and the associated susceptibility are given by

[picture] | (3.28a)

and then

[picture] | (3.28b)

Below and close to the critical temperature , Eq. writes

[picture] | (3.29)

and the solution of this equation is given by

[picture] | (3.30a)

this reduces, for a small surface field , to

[picture] | (3.30b)

and then

[picture] | (3.30c)

The above results, show clearly that the dependence of , and on the shift to critical temperature differs from the behavior of the corresponding bulk properties: while we have , and even remains finite at .

4. Concluding Remarks

We have investigated, in this work, the critical properties at surface within superweak ferrimagnetic materials undergoing a para-ferrimagnetic transition. We have, first, shown that the free energy governing the system can be written as an effective -theory in terms of the overall magnetization The phenomenological parameter multiplying the quadratic and quartic terms are, in this case, functional of the fraction of the magnetization . This leads to an effective extrapolation length appearing in the surface energy term. Through this formulation we have derived, in particular, the overall surface magnetization . We have shown that, in addition of the extrapolation and the bulk correlation length , the surface magnetization depend also on the effective bulk and surface temperature parameters and . We have also determined the surface susceptibilities and , considering the linear response with respect to both the external magnetic field and the local surface field , above and below critical temperature . We have found that the critical exponents remain the same as those relative to the usual -theory, but the amplitudes are changed. We note that a work dealt with the order parameter profiles near surface is in progress.

The mean-field theory (MFT) of critical behavior in the bulk is known to be inaccurate for systems below their marginal dimensionality. There is no reason whatsoever to assume that the MFT for the critical behavior of surfaces is any more accurate than for the bulk. To have non-classical exponents and their associated scaling laws, one has to use appropriate homogeneity assumptions such as finite size scaling theory. Such a work is also in progress.

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