Background dielectric permittivity: material constant or fitting parameter?

TL;DR

This paper critically examines the concept of background dielectric permittivity, arguing that it cannot be consistently defined as a material constant or fitting parameter despite its usefulness in phenomenological models.

Contribution

It provides a critical analysis of the background dielectric constant concept, highlighting its limitations in self-consistent physical modeling.

Findings

Background dielectric constant is useful but not a true material constant.

It cannot be introduced in a self-consistent manner.

The distinction between critical and non-critical polarizations is discussed.

Abstract

The concept of background dielectric constant proposed by Tagantsev, together with the distinction between critical and non-critical electric polarizations as a natural extension for the order parameter of ferroelectric phase transitions, is critically discussed. It is argued and exemplified that, even if these quantities can be very useful for obtaining qualitative and semi-quantitative results from phenomenological modeling, they cannot be introduced in a self-consistent manner.

Equations

$${\mathbf{E}}_{\mathrm{dep}}=-\frac{1}{{\epsilon }_0}\mathbf{P}.$$

$${\mathbf{E}}_{\mathrm{dep}}=-\frac{1}{{\epsilon }_0{\epsilon }_b}{\mathbf{P}}_f.$$

$$F=\frac{\alpha }{2}{P}_f^2+\frac{\beta }{4}{P}_f^4+\frac{1}{2{\epsilon }_0{\chi }_b}{P}_r^2-\left(P_f+P_r\right)E,$$

$$P_f=\frac{1}{{\alpha }^{\prime }(T-T_c)}E,$$

$$P_r={\epsilon }_0{\chi }_{b}E,$$

$$\chi \equiv \frac{\partial P}{\partial E}={\epsilon }_0\left(\frac{C}{T-T_c}+{\chi }_b\right)$$

$$P_{f}=\frac{E}{\alpha^{\prime}\left(T-T_{c}\right)+\alpha^{\prime\prime}\left(T-T_{c}\right)^{2}}=\left(\frac{1}{\alpha^{\prime}\left(T-T_{c}\right)}-\frac{\alpha^{\prime\prime}}{\alpha^{\prime 2}}+\mathcal{O}\big{(}(T-T_{c})\big{)}\right)E,$$

$$\chi=\varepsilon_{0}\left(\frac{C}{T-T_{c}}+\widetilde{\chi}_{b}+\mathcal{O}\big{(}(T-T_{c})\big{)}\right),$$

$$\beta P_f^3+3\gamma P_r{P}_f^2$$

$$\frac{1}{{\epsilon }_0{\chi }_b}{P}_r+\gamma P_f^3$$

$$P_f\approx {\left(E/\beta \right)}^{1/3},$$

$$P_r\approx \left(1-\frac{\gamma }{\beta }\right){\epsilon }_0{\chi }_{b}E.$$

$$\beta P_f^3-3{\gamma }^2{\epsilon }_0{\chi }_b{P}_f^5=\left(1-3\gamma {\epsilon }_0{\chi }_b{P}_f^2\right)E.$$

$$F\left(P_1,P_2;T,E\right)=\frac{a_1}{2}{P}_1^2+\frac{a_2}{2}{P}_2^2+a_{12}{P}_1{P}_2+...-\left(P_1+P_2\right)E.$$

$$a_1{P}_1+a_{12}{P}_2=E,$$

$$a_2{P}_2+a_{12}{P}_1=E.$$

$$P_{1(2)}={\chi }_{1(2)}E,$$

$$P=P_1+P_2=\chi E,$$

$${\chi }_{1(2)}\approx \frac{C_{1(2)}}{T-T_c},$$

$$\chi \approx \frac{C}{T-T_c},$$

$$P=P_1+P_2,$$

$$Q=P_1-P_2.$$

$$F\left(P,Q;T,E\right)=\frac{a_1+a_2+2a_{12}}{8}{P}^2+\frac{a_1+a_2-2a_{12}}{8}{Q}^2+\frac{a_1-a_2}{4}PQ+...-PE$$

$$F\left(P,Q\left(P\right);T,E\right)=\frac{1}{2\chi }{P}^2+...-PE.$$

$$P_f=\frac{1-c}{1+c^2}(P_1-c{P}_2),$$

$$P_r=\frac{1+c}{1+c^2}(c{P}_1+P_2),$$

$$F\left(P_f,P_r;T,E\right)=\frac{A_1^{\prime }\left(T-T_c\right)}{2}{P}_f^2+\frac{A_2}{2}{P}_r^2+A_3^{\prime }\left(T-T_c\right)P_f{P}_r+...-\left(P_f+P_r\right)E,$$

$$A_1^{\prime }=\frac{a_1^{\prime }(T_c)a_2(T_c)+a_2^{\prime }(T_c)a_1(T_c)-2a_{12}^{\prime }(T_c)a_{12}(T_c)}{a_2(T_c){\left(1-c\right)}^2},$$

$$A_2=\frac{a_1^2(T_c)+a_2^2(T_c)+2a_{12}^2(T_c)}{a_2(T_c){\left(1+c\right)}^2},$$

$$A_3^{\prime }=a_{12}^{\prime }(T_c)+\frac{a_1^{\prime }(T_c)-a_2^{\prime }(T_c)}{1-c^2}c.$$

$$A_1^{\prime }\left(T-T_c\right)\leftrightarrow \alpha ,A_2\leftrightarrow \frac{1}{{\epsilon }_0{\chi }_b}.$$

Full text

1 Introduction

The so-called depolarizing field is at the heart of many specific properties of ferroelectric materials. This field is due to (longitudinal) spatial variations of the polarization itself, rather than due to an external source. The key role of the depolarizing field in the formation of ferroelectric domains, for example, was already discussed in the early papers on ferroelectrics (see e.g. [1, 2]) and thanks to a number of complementary studies is rather well understood at present (see e.g. [3, 4]). The impact of the depolarizing field on the fluctuations of the polarization and its relaxation to equilibrium has also been studied by many authors (see e.g. [5]). In this respect, one of the most spectacular effects experimentally verifiably was obtained for the relaxation time of the inhomogenous distribution of polarization in uniaxial ferroelectrics. This distribution can be created by means an acoustic wave, and the attenuation coefficient was predicted to depend on the propagation direction of this wave [6]. Such a dependence was subsequently observed in TGS, but with a dramatic quantitative disagreement with the theoretical prediction [7]. This disagreement was interestingly explained by Tagantsev [8] by separating the total polarization into “order-parameter” (ferroelectric) and “background” (non-ferroelectric) contributions to the total polarization, and further characterizing the latter by means of a “background” (or “base”) dielectric constant. Thus, the difference between theory and experiment was resolved within a Landau-like framework by fitting the value of the background dielectric constant.

In view of this remarkable success, the development of a Landau theory in which the status of the background dielectric constant is upgraded from mere fitting parameter to true material constant has an obvious appeal. This development has been advocated in [9] by invoking some general physical arguments. At first glance, these arguments seem rather natural and convincing. It is clear that, when it comes to its symmetry properties, the order parameter of a proper ferroelectric transition is equivalent to the electric polarization. At the same time, it is also clear that the total polarization generally contains multiple contributions either ionic, electronic, or both, and that only one specific pattern of them can be identified to the actual order parameter (i.e. only one specific pattern is behind the ferroelectric instability). In consequence, together with the “critical” polarization, there are many “non-critical” polarizations that respond to the presence of electric fields but do not emerge spontaneously right after the phase transition. According to [9], the background dielectric constant could be considered as a new material parameter characterizing the non-critical contributions to the total polarization. However, as we show below, this viewpoint has strong conceptual limitations. The Landau theory is in fact quantitatively correct for describing asymptotic behaviors at second-order phase transitions (with the reservations mentioned in Ref. [10]). We shall argue that the incorporation of a non-ferroelectric polarization spoils the internal consistency of the Landau theory of phase transitions and, in consequence, makes it impossible to quantify the background dielectric constant from experimental measurements or first-principle calculations.

This limitation does not mean that the theoretical framework proposed in [9] is impractical. In fact, there are cases in which the non-ferroelectric contributions to the total polarization play an important role. The failure is simply that these contributions cannot be incorporated in a self-consistent way**.** In other words, Tagantsev’s approach cannot be considered as a rightful extension of the Landau theory of phase transitions, but rather as a Landau-like phenomenological modeling providing qualitative or semi-quantitative results only. Beyond that, we recognize that a very important merit of [9] is to stimulate the discussion about the physical meaning of order parameter for proper ferroelectrics. This conceptual discussion is crucial, not only to clarify and to advance the state-of-the-art in phenomenological modeling, but also to build bridges with complementary approaches, notably first-principle-based descriptions. This paper is expected to contribute constructively to this discussion.

The paper is organized as follows. In Sec.2 we introduce the background dielectric constant proposed in [8] by analyzing the depolarizing field created in a ferroelectric slab. In Sec.3 we critically analyze a Landau-like free energy proposed in Ref.[9]. In Sec.4 we discuss the limits of the Landau theory [10] using a fairly general phenomenological model. In Sec.5 we summarize the conceptual results of the paper.

2 Preliminaries

The concept of background dielectric constant can be easily understood by considering the depolarizing field in a slab of an uniaxial ferroelectric.

Consider the simplest case of homogeneous out-of-plane polarization, $\mathbf{P}=P\hat{\mathbf{z}}$, in which the depolarizing field is generated by the corresponding bound charges at the interfaces. This field can be obtained from the electrostatic equation $\nabla\cdot\mathbf{D}=0$ that must be satisfied inside the ferroelectric. Here $\mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}$ is the electric displacement vector, where $\mathbf{E}=\mathbf{E}_{\mathrm{dep}}$ in our case. The symmetry of the problem is such that $D_{x}=D_{y}=0$ and $D_{z}=0$ due to the boundary conditions. Consequently, the depolarizing field is

[TABLE]

The key question raised in [8] can be reformulated in the following way: Can $\mathbf{E}_{\mathrm{dep}}$ be computed within the Landau theory of phase transitions, i.e., from the corresponding order parameter?

The order parameter represents the variable with respect to which the paraelectric phase losses its stability in a second-order transition. In view that it is polar by symmetry, Ginzburg originally identified this variable with the total polarization without much additional justification [11]. It turns out that, sufficiently close to the paraelectric-ferroelectric transition this identification is always possible even though in any real system there are multiple partial polarizations (due to different ions and contributions of electronic clouds) that, from the symmetry point of view**,** are equally good to be the order parameter. Physically, the order parameter corresponds to a specific linear combination of partial polarizations. However, the knowledge of this linear combination is not necessary for the Landau theory where this is not implied. In any case, the order parameter is only one of the many variables characterizing the polar microstructure of the system, and all these variables contribute to the total polarization. The correctness of Ginzburg’s** identification relies in the ferroelectric instability itself: the instability implies a very large response of the order-parameter variable to infinitesimally small stimuli, i.e. softness, such that the total polarization in an external electric field is dominated by this “critical” contribution in the vicinity of the instability point (for an explicit treatment see Sec. 4.1 below). Similarly, the change of the polar microstructure in a short-circuited sample is also dominated by the change in the order parameter close to the instability. However, there are problems in which the rest of polar variables cannot be neglected as their influence can be rather strong. Let us see if this is the case in the the above problem of a ferroelectric slab with uniform polarization. For this, let us write the total polarization as $\mathbf{P}=\mathbf{P}_{f}+\mathbf{P}_{r}$ where $\mathbf{P}_{f}$ represents the ferroelectric order parameter and $\mathbf{P}_{r}$ encloses the rest of contributions to the total polarization.** In this qualitative and preliminary consideration it is natural to assume that $\mathbf{P}_{r}$ is analogous to the regular polarization of a standard (non-ferroelectric) dielectric. In particular, its dependence on the electric field is proposed to be $\mathbf{P}_{r}=\varepsilon_{0}(\varepsilon_{b}-1)\mathbf{E}$ where $\varepsilon_{b}$ is the background dielectric constant [9]. Substituting this into Eq. (1) we obtain:

[TABLE]

If $\varepsilon_{b}$ is analogous to dielectric constant of a standard dielectric, its typical value may be several tens. This means that when the depolarizing field is expressed in terms of the order parameter, a strong influence of the** **regular degrees of freedom can indeed be expected.

Since this situation can be encountered in this and in a number of related problems, the following questions are quite reasonable. Is it possible to develop a theory in which the regular degrees of freedom are included in a consistent way**?** Can the above scheme be more than qualitative, with the background dielectric constant being more than a vaguely defined fitting parameter? These are the questions that we try to answer below.

3 General approach

The description of the ferroelectric transition proposed in Ref. [9] implements the above division between critical and non-critical contributions to the total polarization. Specifically, the standard Landau free energy for an (unixaxial) ferroelectric is replaced by

[TABLE]

where $\chi_{b}=\varepsilon_{b}-1$. Let us check the internal consistency of the proposed framework by working out different examples.

3.1 Temperature dependence of the susceptibility

Let us first consider the linear response of the system in the paraelectric phase. In the case of a temperature-induced second-order transition, it is customary to take $\alpha=\alpha^{\prime}\left(T-T_{c}\right)$ with $T_{c}$ being the transition temperature. According to Eq. (3) we then have

[TABLE]

[TABLE]

and hence

[TABLE]

where $C=1/(\varepsilon_{0}\alpha^{\prime})$. As we see, the Curie-Weiss behavior is obtained from $P_{f}$ while the temperature-independent response is due to $P_{r}$. At first glance, this seems to be a clear-cut way of determining these two polarizations (and hence of measuring the background permittivity). However, if one considers the next order term in $\alpha=\alpha^{\prime}\left(T-T_{c}\right)+\alpha^{\prime\prime}\left(T-T_{c}\right)^{2}$, the critical polarization becomes

[TABLE]

and consequently

[TABLE]

where $\widetilde{\chi}_{b}=\chi_{b}-{\frac{\alpha^{\prime\prime}}{\varepsilon_{0}\alpha^{\prime 2}}}$. This means that, strictly speaking, the temperature-independent part of the total response contains contributions from both $P_{f}$ and $P_{r}$ and, consequently, its connection to the background dielectric constant is ambiguous. Or, put in more constructive terms, the identification of the background dielectric constant with the temperature-independent part of the electric susceptibility is subjected to the condition that the \mathcal{O}\big{(}(T-T_{c})^{2}\big{)} terms in $\alpha$ can be neglected.

3.2 Nonlinear couplings & susceptibility

Another shortcoming is that $P_{r}$ contains corrections due to the presence of high-order terms that are eventually indistinguishable from the obtained from $\varepsilon_{b}$ itself in Eq. (5). In fact the quantities $P_{f}$ and $P_{r}$ have the same transformation properties, and therefore the generalized free energy (3) can contain additional coupling terms. At $T=T_{c}$ the bi-linear coupling $P_{f}P_{r}$ is absent as argued in [9] and we will demonstrate below. However, there is no reason to expect the absence of high-order terms even at $T=T_{c}$. So let us consider these extra terms and address the following question: is it possible to single out critical and non-critical polarizations by studying the electric-field response at $T=T_{c}$?

Consider first the coupling $\gamma P_{r}P_{f}^{3}.$ Including this coupling into the free energy Eq. (3) and further minimizing with respect to $P_{f}$ and $P_{r}$ we obtain

[TABLE]

If the external field is sufficiently low this givies

[TABLE]

[TABLE]

By comparing Eqs. (12) and (5) we can see that the high-order coupling in fact generates a correction to $P_{r}$ that is also linear in $E$. Should we take it into account? If yes, then not only it. In fact, combining Eqs. (9) and (10) one obtains

[TABLE]

In this equation there appears a $P_{f}^{5}$ term that plays the same role as the obtained from a sixth-order $P_{f}^{6}$ term in the generalized free energy. This means that, if we include $P_{r}$ in Eq. (3), then we have to do it by introducing not only the $P_{r}^{2}$ term in the initial free energy, but also the $P_{f}^{3}P_{r}$ and $P_{f}^{6}$ terms that essentially play the same role. Or in other words, the field dependence of $P_{f}$ and $P_{r}$ turns out to be determined not only by the background dielectric constant $\varepsilon_{b}$, but also by the coefficients of these high-order terms. The incorporation of all these terms does not sound very practical, and the extended framework thus losses its appeal.

We recall that the internal consistency of the Landau theory lies on the fact that only asymptotic behaviors near the transition point are considered. In this sense, Eq. (11) provides such an asymptotic behavior while Eq. (12) is beyond the scope of the Landau theory since ${E\to 0}\mathrm{lim}{P_{r}/P_{f}}=0$.

3.3 Spontaneous polarization

Another example is temperature dependence of the total spontaneous polarization ($E=0$). Can we individuate its critical and non-critical contributions? According to Eq. (3) $P_{r}=0$ at zero field, but taking into account the coupling $\gamma P_{r}P_{f}^{3}$ one obtains $P_{r}\propto P_{f}^{3}\propto(T-T_{c})^{3/2}$. However, the same dependence is obtained from the bilinear coupling which, as we will see the next section, has the form $\alpha_{12}\left(T-T_{c}\right)P_{f}P_{r}$. In addition, the terms \mathcal{O}\big{(}(T-T_{c})^{2}\big{)} and $\mathcal{O}(T-T_{c})$ in the Taylor series expansion of the coefficients $\alpha$ and $\beta$ respectively also result into a similar** $(T-T_{c})^{3/2}$ dependence of $P_{f}$**. Consequently, we see once again that $P_{r}$ and $P_{f}$ are poorly defined quantities with no direct experimental access: even if the subdominant $\left(T-T_{c}\right)^{3/2}$ contribution to the total spontaneous polarization is determined experimentally, one cannot conclusively say if this is due to the lowest-order contribution to $P_{r}$, high-order contributions to $P_{f}$, or both.

3.4 Inhomogeneous depolarizing fields

The notion of background dielectric constant plays its most important role in problems related to* homogeneous or almost homogeneous depolarizing field. There are many problems, however, where this field is strongly inhomogeneous. A typical example is formation of domain structure in not extremely thin films. Here the scale of changes in the depolarizing field is much less than the film thickness. problem of the . In this case the response of the system perpendicular to the ferroelectric axis comes to the scene on top of the non-critical response along the ferroelectric direction. This is because an inhomogenous depolarizing field necessarily has to have multiple components according to the electrostatic equation $\nabla\times\mathbf{E}=0$. As a result, $\varepsilon_{b}$ is generally in competition with well-defined material constants such as the dielectric constant perpendicular to the ferroelectric axis. The good news is that, even if the results obtained from the Landau-like formalism nominally depend on $\varepsilon_{b}$, they can nonetheless be dominated by true material constants and hence be robust results. An example of this situation which  can be found in Ref.[12] where the Landau-like approach was used to interpret * experimental data of Ref.[13]. Strong effect of film thickness ($l$) on ferroelectric phase transition temperature ($T_{c}$) in non-electroded BaTiO3 films.on SrTiO3 substrate was revealed in this paper: at changing the film thickness from $1.6\unit{nm}$ to $10\unit{nm}$ the phase transition temperature changed from $70\unit{K}$ to $925\unit{K}.$Theoretically this phase transition is expected to be into multidomain state .The only parameter of the Landau-like approach unknown from independent experiments was $\varepsilon_{b}$. It proved out that changing of $\varepsilon_{b}$ from $1$ to $10$ is almost unnoticeable for the theoretical $T_{c}\left(l\right)$ curve while its changing from $10$ to $100$ has small but still noticeable effect for $l<5\unit{nm}.$

4 Model approach

In order to further clarify the physical meaning of the variables introduced in [9], it is instructive to reconsider the loss of stability of the paraelectric phase from the perspective of a specific model and its effective Hamiltonian (or incomplete free energy). For this, let us consider the simplest situation in which we have just two “microscopic” contributions to the total polarization, $P=P_{1}+P_{2}$, and the most general effective Hamiltonian allowed by symmetry:

[TABLE]

The individual polarizations $P_{1}$ and $P_{2}$ can be, for example, the ionic polarizations of two different atoms or the ionic and electronic polarizations of the same atom. For the sake of concreteness we consider the temperature as the control parameter. Accordingly, the coefficients of this Hamiltonian are assumed to be (unknown) functions of this parameter.

4.1 Individual vs. total polarization

The loss of stability of the paraelectric phase can be analyzed from the linearized equations of state (or equilibrium equations):

[TABLE]

[TABLE]

The stability of the paraelectric phase ($P_{1}=P_{2}=0$) as the ground state of the system requires $a_{1}$, $a_{2}$, and $a_{1}a_{2}-a_{12}^{2}$ to be $>0$. Conversely, the paraelectric phase losses its stability at the point at which $a_{1}a_{2}-a_{12}^{2}=0$. Since this point can be reached from the paralectric phase where $a_{1}$ and $a_{2}$ are $>0$, this means that, in general, both $a_{1}$ and $a_{2}$ are positive at the transition point (unless $a_{12}=0$). In addition, the sign of $a_{12}$ determines the precise state that emerges in the transition: $P_{1}$ and $P_{2}$ have the same sign (parallel) if $a_{12}<0$, while they have opposite sign (anti-parallel) if $a_{12}>0$. In the following we consider $a_{12}<0$, although essentially the same is obtained for $a_{12}<0$.

Near the transition point it is generically possible to define the Taylor expansion $a_{1}a_{2}-a_{12}^{2}\equiv\Delta=\Delta^{\prime}(T-T_{c})+\dots$, where $T_{c}$ is the transition temperature and $\Delta^{\prime}$ a positive constant (such that $a_{1}a_{2}-a_{12}^{2}>0$ for $T>T_{c}$). In the paraelectric phase, this automatically gives the Curie-Weiss-law behavior for both individual and total polarizations:

[TABLE]

[TABLE]

where, according to Eqs. (15) and (16),

[TABLE]

[TABLE]

with $C_{1(2)}=[a_{2(1)}\left(T_{c}\right)-a_{12}\left(T_{c}\right)]/\Delta^{\prime}$ and $C=[a_{1}\left(T_{c}\right)+a_{2}\left(T_{c}\right)-2a_{12}\left(T_{c}\right)]/\Delta^{\prime}$. Note that all these susceptibilities are positive since $a_{1(2)}(T_{c})>0$ and $a_{12}(T_{c})<0$.

As we see, all these polarizations display the same divergent behavior close the transition point and, consequently, all of them can be considered as critical (or, conversely, none of them is non-critical). Accordingly, the order parameter of the transition can be associated to either of them. In this sense, the total polarization is a rather natural choice as Ginzburg originally considered in [11]. However, to obtain Ginzburg’s free energy, one still has to minimize (or integrate out) over all the variables that do not contribute to the total polarization. In the case of the effective Hamiltonian (14), this can be done by changing to the variables

[TABLE]

[TABLE]

Thus, in terms of these variables the Hamiltonian reads

[TABLE]

and the minimization over $Q$ yields

[TABLE]

The functional obtained in this way corresponds to the standard free energy in the Landau theory of phase transitions.

4.2 Critical and non-critical polarizations

In order to individuate critical and non-critical polarizations in our model (and hence the background dielectric constant) we have to proceed differently. The key point is to identify the actual degree of freedom (or generalized coordinate) with respect to which the paraelectric phase becomes unstable. That is, the degree of freedom describing the structural changes that will emerge spontaneously right after the phase transition. Mathematically, this corresponds to the eigenvector of the quadratic form in Eq. (14) whose eigenvalue vanishes and hence defines the transition point. By performing standard linear algebra, one can easily find that the linear combinations

[TABLE]

[TABLE]

with $c=\frac{a_{1}\left(T_{c}\right)}{a_{12}\left(T_{c}\right)}=\frac{a_{12}\left(T_{c}\right)}{a_{2}\left(T_{c}\right)}$, represent the eigenvectors of the model at $T_{c}$. Accordingly, the effective Hamiltonian can be written as

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

to the lowest relevant order in $T-T_{c}$. In this model, $P_{f}$ is therefore the critical polarization while $P_{r}$ is the non-critical one. In fact, the effective Hamiltonian can be related to Eq. (3) by identifying

[TABLE]

Compared to Eq. (3), however, Eq. (27) contains an additional coupling term $P_{f}P_{r}$. This coupling vanishes at $T=T_{c}$ since, by construction, only $P_{f}$ is behind the stability. At $T\not=T_{c}$, in contrast, the coupling is nonzero, which simply tells us that the physical content of the variables diagonalizing the Hamiltonian is different at different temperatures. In Ref. [9], however, this coupling was put to zero also for $T\not=T_{c}$, which does not have an obvious justification.

We find it instructive to discuss the possible origin of this subtle mistake. In fact, we feel that two of the authors might have been reponsible for it, as they made the same before Ref. [9] and in Refs. [14],[15]. It traces back to the tacit assumption that the order parameter can be directly identified with a microscopic variable. Strictly speaking, the Landau free energy is formally obtained by i) identifying the complete set of microscopic variables with the same transformation properties ii) integrating out over the rest, iii) choosing the appropriate linear combination of the remaining variables and iv) minimizing over the rest. The choice in step iii) is determined by physical considerations related to the specific problem under consideration. And in step iv) integration can be replaced by a simple minimization provided that the corresponding degrees of freedom have measure zero compared to the total degrees of freedom. But, beyond this, one essential point to be kept in mind that, from step ii), one is dealing with an effective Hamiltonian whose parameters, unlike in the initial Hamiltonian, depend on the corresponding control parameters (e.g. temperature or pressure). In consequence, its diagonalization is in general different for different values of the these control parameters. As we have seen, the latter plays a role when critical and non-critical order parameters are considered beyond the standard Landau theory of phase transitions.

5 Conclusions

We have illustrated that the generality, consistency, and the model-independency of the Landau theory of phase transitions is not automatically inherited by Landau-like phenomenological models, even when they logically seek to extend the perimeter of action of the Landau theory. By considering the ferroelectric case, we have demonstrated that the incorporation of non-critical polarizations and the corresponding background dielectric constant is possible, but at the expense of sacrificing the full internal consistency of the theory. The background dielectric constant, in particular, turns out to be a mere fitting parameter –and not a true material constant in its own right. This parameter still retains a more or less precise physical meaning. In consequence, some physical intuition is needed to determine the range of acceptable values in a given problem and, beyond that, the results must be robust within this range for them to be trustable.

We thank A. K. Tagantsev for vivid and stimulating discussions.