Quantum Theory of Spontaneous Polarization in a Ferroelectric Materials

1. Introduction

Segnetoelectrics are materials that typically possess a crystalline structure characterized by ionic bonds and a unit cell lacking a center of symmetry (which makes them all piezoelectrics); they also feature polar axes of symmetry. Certain nanostructured and composite materials, liquid crystals, ceramics, and specially prepared polymers can also exhibit pyroelectric properties. Ferroelectric PbHfO₃–PbSnO₃ solid solutions with a perovskite structure serve as an example of the formation of an incommensurate phase. This process is driven by the emergence of an intermediate phase—situated between the cubic and incommensurate phases—which is stabilized in PbHfO₃ upon the addition of PbSnO₃. It has been experimentally determined that, within the cubic phase, the ferroelectric stiffness exhibits a linear temperature dependence, analogous to the Curie–Weiss law [1]. To describe the experimental results, a model based on Landau’s phenomenological theory is employed, incorporating two coupled order parameters: a ferroelectric order parameter and an antiferrodistortive order parameter. While phenomenological theories, such as the Curie-Weiss theory, can to some extent describe macroscopic manifestations—including the polarization vector (the macroscopic dipole moment) and the associated hysteresis—they fail to link these phenomena to the underlying microscopic ordering of individual dipole moments. In other words, while the phenomenological thermodynamic theory demonstrates the connection between various phenomena in ferroelectric crystals, it cannot explain the Curie-Weiss law—that is, it cannot determine the origin of spontaneous polarization. This is the task of a microscopic, molecular theory of ferroelectricity. Despite the significant importance of these materials for applications in microelectronics, there is no comprehensive theory that explains, at a microscopic level, the very origin of spontaneous polarization in ferroelectrics. Moreover, unlike ferromagnets—for which there exist models based on the Hamiltonian derived by Heisenberg—there is no consistent microscopic model for ferroelectrics that addresses the fundamental nature and origin of this collective phenomenon. This phenomenon involves the alignment of dipoles, which results in the formation of a collective, aggregate dipole moment. The attempts to explain the dipole ordering in ferroelectric based on the idea of direct dipole-dipole (DD) interaction provokes many questions, so, from the electrostatics point of view, this interaction makes an antiparallel ordering of dipole coupling, that could interpreters an anti-ferroelectric ordering. A similar attempt was made in [2] a study focused on PbZrO3 crystals, aiming to determine the origin of their antiferroelectric ordering. However, the notion that dipole-dipole (DD) interactions can promote non-uniform ordering is questionable—particularly in cases where one assumes a high total polarizability for A-type sites, a comparable total polarizability for oxygen sites within A–O planes, and a relatively weak total polarizability for oxygen sites along the B–O direction. Direct dipole-dipole interaction alone cannot account for such dipole behavior within the crystal.was done in the work [2]. Meanwhile, in atomic physics, there are other analogous collective phenomena involving the coherent ordering of dipole moments that possess a clear quantum-mechanical interpretation—one that has proven highly successful in explaining their general observable characteristics. A prime example is the Dicke theory [3]of superradiance; its core feature is the summation of dipole moments, which leads to the emergence of a collective macroscopic dipole moment. This, in turn, results in an emission intensity that is proportional to the square of the number of atoms (or dipole moments), i.e., N². Phase transitions in quantum-optical systems are also driven by interactions that lead to the establishment of a specific order in the orientation of so-called isoenergetic spins—a phenomenon observed in Dicke superradiance [3]. Furthermore, an analysis of the state of a polariton laser in an open medium—comprising two-level atoms that interact via a Stokes field during Raman scattering—has demonstrated that the superradiant regime represents one possible type of phase transition. [4] However, the mechanical transfer of the Heisenberg Hamiltonian form from the theory of ferromagnetism to the theory of ferroelectrics—a practice adopted by some authors [5,6]—is not always justified. This is because the SU(2) rotation group (to which objects with spin 1/2 belong) and its characteristic commutation relations for spin projection operators—defined with respect to a Cartesian or any other orthogonal basis—do not hold for the projection operators of a dipole moment along the same axes. In any case, one would need to specifically prove that dipole moments actually belong to this particular rotation group. Furthermore, there remains the alternative possibility that they belong to other types of three-dimensional rotation groups, such as SO(3) or R³. Nevertheless, it was really correct attempt to account a cooperative couplings between octahedral rotations and ferroelectricity in perovskites [7].

In our work, we propose a quantum model of rotating dipoles, which allows for the interpretation of the formation of a collective dipole moment within a ferroelectric domain; this model is founded on fundamental symmetry principles, including the rotational symmetry of the wave functions that describe the quantum state of the electron cloud within a crystal unit cell.

2. Rotating Dipole in the Symmetric Model

The piezoelectric effect occurs in ionic crystals. It arises from a change in the relative positions of ions due to deformation, which results in the generation of an electric dipole moment. At the macroscopic level, it has been demonstrated—based on the Fumi method—that the piezoelectric effect can only occur in crystals that lack a center of symmetry. For instance, the crystalline modification of barium titanate with a perovskite structure is a ferroelectric material that exhibits both photorefractive and piezoelectric effects. As the temperature decreases, barium titanate crystals undergo a series of successive ferroelectric phase transitions: at 120 °C, they transition from a cubic (paraelectric) phase to a tetragonal polar (ferroelectric) phase; this is followed at 5 °C by a transition to an orthorhombic polar phase; and finally, at −90 °C, the material transitions into a rhombohedral polar phase. All three are first-order phase transitions, meaning that the dielectric constant changes discontinuously as a function of temperature. Above the Curie temperature Tc of 120 °C, the dielectric constant follows the Curie–Weiss law. Considering the structure of the unit cell—using barium titanate as an example—in which a tetravalent titanium ion (Ti⁴⁺) is positioned at the center of a cubic lattice, while negatively charged oxygen ions (O²⁻) occupy the centers of the cube's six faces and divalent barium ions (Ba²⁺) reside at the eight vertices, it can be argued that the electron density is primarily "drawn" toward the oxygen ions, yet it retains a degree of mobility within the cell. This specific condition enables the formation of a non-zero electric dipole moment within the unit cell. Since the emergence of this dipole moment is linked to the displacement of the central Ti⁴⁺ ion—which breaks the central symmetry—the velocity of this ion’s displacement (along with the associated mobile electron cloud "pulled" toward the oxygen ions) induces a slight rotation of the entire dipole relative to the original center of symmetry. We shall adopt a model of a dipole moment that undergoes slow rotation within a complex crystalline field, analogous to the rotation of a polar diatomic molecule characterized by an ionic type of interatomic bonding. Consequently, for the purpose of further calculations, we will represent the dipole within the unit cell as a quantum rotator with low values of angular momentum. A quantum rotator is a model of a microscopic object that possesses no internal or external properties other than a specific mass distribution. This mass distribution is described by an inertia tensor. The motion of this object involves rotations about coordinate axes and is quantized, in the sense that the constants of motion—namely, rotational energy and angular momentum—can take on only specific, discrete values. The set of allowed energy values constitutes the energy spectrum of the rotator, while the rotations themselves are described by stationary wave functions. Quantum rotators serve as models for the rotational properties of molecules and atomic nuclei, enabling the study of their rotational spectra. Without loss of generality, we shall assume that the axes of the body-fixed—or "molecular"—coordinate system coincide with the principal axes of the inertia tensor.

In the general case, the Hamiltonian of an asymmetric rotator has the form:

$$\begin{array}{l}\widehat{H}=A{\widehat{J}}_a^C{\widehat{J}}_a^C+B{\widehat{J}}_b^C{\widehat{J}}_b^C+C{\widehat{J}}_c^C{\widehat{J}}_c^C,\\ A={\displaystyle \frac{{\hslash }^2}{2I_a}},B={\displaystyle \frac{{\hslash }^2}{2I_b}},C={\displaystyle \frac{{\hslash }^2}{2I_c}},\end{array}$$

(1)

Here, the quantities A, B, and C are called rotational constants, the operators ${\widehat{J}}_a^C,{\widehat{J}}_b^C,{\widehat{J}}_c^C$ represent the projections of the angular momentum onto the axes of the molecule-fixed coordinate system (in the center-of-mass frame), and the indices a, b, and c can take the values 1, 2, 3 or x, y, z. It should also be noted that the factor ℏ² is incorporated into the rotational constants themselves; therefore, we will not explicitly include it when writing out the angular momentum projections or the energy expressions. In accordance with standard practice (see, for example, [8]), the convention used for the rotational constants—where A is associated with the smallest principal moment of inertia and C with the largest—is a widely accepted spectroscopic convention adopted for convenience. In our case, we will examine the energy spectrum and wave functions of quantum rotators, specifically for symmetric and spherical rotators. The Hamiltonian for a symmetric rotator is defined by the following formulas for an oblate (denoted by the subscript ‘f’ — "flattened") and a prolate (denoted by the subscript ‘e’ — "elongated") rotator, respectively:

$$\begin{array}{l}\mathit{Li}{m}_{A\to B}\widehat{H}=H_{\mathit{Sf}}=B{\widehat{J}}^2+\left(C-B\right){\widehat{J}}_c^C{\widehat{J}}_c^C,\\ \mathit{Li}{m}_{C\to B}\widehat{H}=H_{\mathit{Se}}=B{\widehat{J}}^2+\left(A-B\right){\widehat{J}}_a^C{\widehat{J}}_a^C,\end{array}$$

(2)

The energy spectrum of a quantum rotator consists of the eigenvalues of the Hamiltonian, and the wave functions are the eigenfunctions of the Hamiltonian. Furthermore, the wave functions are eigenfunctions of the operators for the square of the angular momentum ${\widehat{J}}^2=\widehat{J}\cdot \widehat{J}$, and its projections ${\widehat{J}}_Z^L$ onto the Z-axis of the laboratory coordinate system (LCS) and ${\widehat{J}}_z^C$ the z-axis of the molecular coordinate system (MCS). Given that the Hamiltonian can also be expressed in terms of the square of the total angular momentum, it follows that the functions which are simultaneous eigenfunctions of all these operators must depend on at least three quantum numbers: the rotational quantum number J (often referred to simply as the angular momentum), the quantum number K representing the projection of the angular momentum onto the z-axis (simply called the projection onto the quantization axis in the molecule-fixed coordinate system, MFCS), and the quantum number M representing the projection of the angular momentum onto the Z-axis (also called the projection onto the quantization axis in the laboratory-fixed coordinate system, LFCS). We shall denote this function asψ_JMK. Thus, for an oblate symmetric top (A = B), we have:

$$\begin{array}{l}{H}_{\mathit{Sf}}=B{\widehat{J}}^2+\left(C-B\right){\widehat{J}}_c^C{\widehat{J}}_c^C,\\ {\widehat{J}}^2{\psi }_{\mathit{JMK}}=J(J+1){\psi }_{\mathit{JMK}},J=0,1,2,\dots ,\\ {\widehat{J}}_z^C{\psi }_{\mathit{JMK}}=K{\psi }_{\mathit{JMK}},K=-J,-J+1,\dots 0,\dots J-1,J,\\ {\widehat{J}}_Z^L{\psi }_{\mathit{JMK}}=M{\psi }_{\mathit{JMK}},M=-J,-J+1,\mathrm{...0},\dots J-1,J,\\ H_{\mathit{Sf}}{\psi }_{\mathit{JMK}}=E_{\mathit{JMK}}{\psi }_{\mathit{JMK}}.\end{array}$$

(3)

where the quantities are the eigenvalues of the Hamiltonian, which represent the energy spectrum of the rotator. Let us note an important fact regarding this energy spectrum: since the Hamiltonian does not depend explicitly on derivatives with respect to the angle α—which describes rotation about the Z-axis of the laboratory coordinate system—the energy spectrum of the rotator is independent of the quantum number M; in other words, the rotational energy is degenerate with respect to this quantum number. A function satisfying these equations is the Wigner D-function (see Chapter 4 [8]) $D_{\mathit{KM}}^J\left(\mathit{\alpha \beta \gamma }\right)$, which is normalized such that the integral of the squared modulus of the rotator's wave function over the entire range of angles equals unity. The angles α,β,γ are uniquely related to the Euler angles by the relations $\varphi =\alpha +{\displaystyle \frac{\pi }{2}},\theta =\beta ,\chi =\gamma -{\displaystyle \frac{\pi }{2}}$ [6]; here, the angles ϕ,θ,χare the Euler angles. Thus, the wave function of the rotator takes the following form:

$$\begin{array}{l}{\psi }_{\mathit{JMK}}(\mathit{\alpha \beta \gamma })\equiv \left|\mathit{JMK}\right.〉=\sqrt{{\displaystyle \frac{2J+1}{8{\pi }^2}}}{D}_{\mathit{MK}}^J\left(\alpha ,\beta ,\gamma \right)=\\ =i^{K-M}\sqrt{{\displaystyle \frac{2J+1}{8{\pi }^2}}}{d}_{\mathit{MK}}\left(\theta \right)e^{\mathit{iM\varphi }}{e}^{\mathit{iK\chi }}.\end{array}$$

(4)

The function $d_{\mathit{MK}}\left(\theta \right)$ in the form of finite trigonometric sums (see [8] ). The energy spectrum of a flattened symmetric rotator will be represented in the following form:

E_JMK = BJ(J + 1) − (B − C)K².

(5)

For an elongated rotator (B=C) we have following:

$$\begin{array}{l}{\widehat{H}}_{\mathit{Se}}{\psi }_{\mathit{JMK}}=E_{\mathit{JMK}}{\psi }_{\mathit{JMK}},\\ {\widehat{H}}_{\mathit{Se}}=B{\widehat{J}}^2+\left(A-B\right){\widehat{J}}_a^C{\widehat{J}}_a^C,\\ E_{\mathit{JMK}}=\mathit{BJ}(J+1)+(A-B)K^2.\end{array}$$

(6)

The energy spectrum of a symmetric top exhibits a twofold degeneracy of rotational levels with respect to the quantum number K; that is, the rotational energy takes the same value for two mutually opposite directions of the angular momentum relative to the top's symmetry axis. Technically, this arises because the Hamiltonian depends only on the second derivative with respect to the angle and is invariant under reflections in a plane passing through the symmetry axis of the top. As long as we are concerned solely with the pure rotational spectra of a symmetric top, this K-degeneracy cannot be lifted, and the wave functions (4) provide an adequate description of the quantum states. However, if the motion of the top is perturbed in any way, this degeneracy may be lifted, and the energy levels will then depend on the sign of the quantum number K.

Let us calculate the matrix elements of the electric dipole moment for the rotational states of a symmetric top (symmetric rotator). Due to the symmetry of the rotator, the dipole moment vector is directed along its symmetry axis; thus, the only non-zero component of this vector is its projection onto the Z' axis of the body-fixed coordinate system, which is equal to the magnitude of the vector itself. The quantities ${\widehat{D}}_{\mu }\left(\alpha ,\beta ,\gamma \right)$(μ=+1, -1, 0 ) are the cyclic components (projections) of the dipole moment operator onto the axes of the laboratory-fixed coordinate system (LFCS). Here, we do not consider the internal structure of the rotator; the operators for the dipole moment components of the rotator are assumed to be defined by their projections onto the axes of the molecule-fixed coordinate system (MFCS).In other words, in the formula describing the cyclic components (projections) of the dipole moment operator along the axes of the laboratory reference frame, only the term with ν=0 remains. These dipole moment matrix elements represent the quantum-mechanical averages of the dipole moment—arising from the electron distribution in the states under consideration—within the crystalline field of a unit cell. Since the projections of the dipole moment constitute the components of an irreducible tensor of the first rank, the relationships between these projections ${\widehat{D}}_{\mu }\left(\alpha ,\beta ,\gamma \right)$ and ${\widehat{D}}_{\nu }\left(\alpha ,\beta ,\gamma \right)$ determined by

$${\widehat{D}}_{\mu }\left(\alpha ,\beta ,\gamma \right)={\displaystyle \sum _{\nu =0,\pm 1}{D}_{\mathit{\mu \nu }}^{1*}\left(\alpha ,\beta ,\gamma \right)}\widehat{D}{\text{'}}_{\nu }\left(\alpha ,\beta ,\gamma \right)$$

(7)

Here $D_{\mathit{\mu \nu }}^{1*}\left(\alpha ,\beta ,\gamma \right)$ is a Wigner function. Therefore, in the atom units (e=1, a_B=1, Δ- is linear size of dipole in Bohr radius a_B )

$$\begin{array}{l}\left.〈J_f{M}_f{K}_f\right|{\widehat{D}}_{\mu }\left|J_i{M}_i{K}_i\right.〉=i^{M_i-M_f-K_i+K_f}{\displaystyle \frac{\sqrt{\left(2J_i+1\right)\left(2J_f+1\right)}}{8{\pi }^2}}\times \\ \times {\displaystyle {\displaystyle \underset{0}{\overset{2\pi }{\int }}}{\displaystyle {\displaystyle \underset{0}{\overset{2\pi }{\int }}}{\displaystyle {\displaystyle \underset{0}{\overset{2\pi }{\int }}}{D}_{M_f{K}_f}^{J_f}\left(\alpha ,\beta ,\gamma \right)D_{\mu 0}^{1*}\left(\alpha ,\beta ,\gamma \right)D_{M_i{K}_i}^{J_i*}\left(\alpha ,\beta ,\gamma \right)\mathit{sin\beta d\gamma d\beta d\alpha }}}}=\\ =\Delta i^{\mu +M_i-M_f}\sqrt{{\displaystyle \frac{\left(2J_i+1\right)}{\left(2J_f+1\right)}}}{C}_{J_i{K}_{i}10}^{J_f{K}_f}{C}_{J_i{M}_{i}1\mu }^{J_f{M}_f}.\end{array}$$

(8)

Let us rewrite this result in the form of the Wigner-Eckart theorem:

$$\begin{array}{l}\left.〈J_f{M}_f{K}_f\right|{\widehat{D}}_{\mu }\left|J_i{M}_i{K}_i\right.〉={\displaystyle \frac{i^{M_i-M_f+\mu }}{\sqrt{\left(2J_f+1\right)}}}{C}_{J_i{M}_{i}1\mu }^{J_f{M}_f}\left.〈J_f{K}_f\right|\left|{\widehat{D}}_{\mu }\right|\left|J_i{K}_i\right.〉,\\ \left.〈J_f{K}_f\right|\left|{\widehat{D}}_{\mu }\right|\left|J_i{K}_i\right.〉=\Delta \sqrt{\left(2J_i+1\right)}{C}_{J_i{K}_{i}10}^{J_f{K}_f}{\delta }_{K_f{K}_i}\end{array}$$

(9)

The selection rules are entirely determined by the Clebsch–Gordan coefficients; that is, for the coefficients to be non-zero, the following conditions must be met:

$$\begin{array}{l}{J}_f=J_i,J_i\pm 1,\\ K_f=K_i\end{array}$$

(10)

Next, let us consider the case of the slowest rotation; taking the electron spin into account, the total angular momentum can reach a minimum value.

$$\begin{array}{l}{J}_f=J_i=1/2,\\ J_f=J_i+1=3/2.\end{array}$$

(11)

We shall examine the case J_f = J_i = 1/2, in detail.

The cyclic component ${\widehat{D}}_{\mu =0}$ of the dipole moment has the following non-zero matrix elements

$$\begin{array}{l}\left.〈J_f{\displaystyle \frac{1}{2}}{M}_f{\displaystyle \frac{1}{2}}{K}_f\right|{\widehat{D}}_{\mu =0}\left|J_i{\displaystyle \frac{1}{2}}{M}_i{\displaystyle \frac{1}{2}}{K}_i\right.〉={\displaystyle \frac{i^0}{\sqrt{\left(2J_f+1\right)}}}{C}_{J_i{\displaystyle \frac{1}{2}}{M}_i{\displaystyle \frac{1}{2}}1\mu 0}^{J_f{\displaystyle \frac{1}{2}}{M}_f{\displaystyle \frac{1}{2}}}\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu =0}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉=\\ ={\displaystyle \frac{i^0}{\sqrt{\left(2\frac{1}{2}+1\right)}}}{C}_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}10}^{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu =0}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left(-{\displaystyle \frac{1}{\sqrt{3}}}\right)\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu =0}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉\\ \left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu =0}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉=\Delta \sqrt{\left(2{\displaystyle \frac{1}{2}}+1\right)}{C}_{J_i{\displaystyle \frac{1}{2}}{K}_i{\displaystyle \frac{1}{2}}10}^{J_f{\displaystyle \frac{1}{2}}{K}_f{\displaystyle \frac{1}{2}}}=\Delta \sqrt{2}{C}_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}10}^{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}=-{\displaystyle \frac{\sqrt{2}}{\sqrt{3}}}\Delta ,\Rightarrow \\ \left.〈J_f={\displaystyle \frac{1}{2}}{M}_f={\displaystyle \frac{1}{2}}{K}_f={\displaystyle \frac{1}{2}}\right|{\widehat{D}}_{\mu =0}\left|J_i={\displaystyle \frac{1}{2}}{M}_i={\displaystyle \frac{1}{2}}{K}_i={\displaystyle \frac{1}{2}}\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left(-{\displaystyle \frac{1}{\sqrt{3}}}\right)\left(-{\displaystyle \frac{\sqrt{2}}{\sqrt{3}}}\Delta \right)={\displaystyle \frac{\Delta }{3}},\\ \left.〈J_f={\displaystyle \frac{1}{2}}{M}_f={\displaystyle \frac{1}{2}}{K}_f=-{\displaystyle \frac{1}{2}}\right|{\widehat{D}}_{\mu =0}\left|J_i={\displaystyle \frac{1}{2}}{M}_i={\displaystyle \frac{1}{2}}{K}_i=-{\displaystyle \frac{1}{2}}\right.〉=-{\displaystyle \frac{\Delta }{3}}.\end{array}$$

(12)

The cyclic component ${\widehat{D}}_{\mu =1}$ of the dipole moment has the following non-zero matrix elements

$$\begin{array}{l}\left.〈J_f={\displaystyle \frac{1}{2}}{M}_f={\displaystyle \frac{1}{2}}{K}_f\right|{\widehat{D}}_{\mu =1}\left|J_i={\displaystyle \frac{1}{2}}{M}_i=-{\displaystyle \frac{1}{2}}{K}_i\right.〉={\displaystyle \frac{i^{-1+1}}{\sqrt{2}}}{C}_{J_i{\displaystyle \frac{1}{2}}{M}_i\left(-{\displaystyle \frac{1}{2}}\right)1\mu 1}^{J_f{\displaystyle \frac{1}{2}}{M}_f{\displaystyle \frac{1}{2}}}\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu 1}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉=\\ ={\displaystyle \frac{i^0}{\sqrt{2}}}{C}_{{\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{1}{2}}\right)11}^{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu 1}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉={\displaystyle \frac{i^0}{\sqrt{2}}}\sqrt{{\displaystyle \frac{2}{3}}}\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu 1}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉=\\ =\sqrt{{\displaystyle \frac{1}{3}}}\sqrt{{\displaystyle \frac{2}{3}}}=-{\displaystyle \frac{\sqrt{2}}{3}}\\ \left.〈J_f{\displaystyle \frac{1}{2}}{K}_f{\displaystyle \frac{1}{2}}\right|\left|{\widehat{D}}_{\mu 1}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i{\displaystyle \frac{1}{2}}\right.〉=\Delta \sqrt{\left(2{\displaystyle \frac{1}{2}}+1\right)}{C}_{J_i{\displaystyle \frac{1}{2}}{K}_i{\displaystyle \frac{1}{2}}10}^{J_f{\displaystyle \frac{1}{2}}{K}_f{\displaystyle \frac{1}{2}}}=\sqrt{2}{C}_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}10}^{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}}\Delta =-\sqrt{{\displaystyle \frac{2}{3}}}\Delta \end{array}$$

(13)

The cyclic component ${\widehat{D}}_{\mu =-1}$ of the dipole moment has the following non-zero matrix elements

$$\begin{array}{l}\left.〈J_f={\displaystyle \frac{1}{2}}{M}_f=-{\displaystyle \frac{1}{2}}{K}_f\right|{\widehat{D}}_{\mu =-1}\left|J_i={\displaystyle \frac{1}{2}}{M}_i={\displaystyle \frac{1}{2}}{K}_i\right.〉={\displaystyle \frac{i^{1+1}}{\sqrt{2}}}{C}_{J_i{\displaystyle \frac{1}{2}}{M}_i{\displaystyle \frac{1}{2}}1\mu \left(-1\right)}^{J_f{\displaystyle \frac{1}{2}}{M}_f-{\displaystyle \frac{1}{2}}}\left.〈J_f={\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu \left(-1\right)}\right|\left|J_i={\displaystyle \frac{1}{2}}{K}_i\right.〉=\\ ={\displaystyle \frac{i^{1+1}}{\sqrt{2}}}{C}_{{\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{2}}1\left(-1\right)}^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}}\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu \left(-1\right)}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉=-{\displaystyle \frac{i^{1+1}}{\sqrt{2}}}\sqrt{{\displaystyle \frac{1}{3}}}\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu \left(-1\right)}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉=\\ =\sqrt{{\displaystyle \frac{1}{3}}}\left.〈J_f{\displaystyle \frac{1}{2}}{K}_f\right|\left|{\widehat{D}}_{\mu \left(-1\right)}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i\right.〉,\\ \left.〈J_f{\displaystyle \frac{1}{2}}{K}_f-{\displaystyle \frac{1}{2}}\right|\left|{\widehat{D}}_{\mu \left(-1\right)}\right|\left|J_i{\displaystyle \frac{1}{2}}{K}_i-{\displaystyle \frac{1}{2}}\right.〉=\sqrt{\left(2{\displaystyle \frac{1}{2}}+1\right)}{C}_{J_i{\displaystyle \frac{1}{2}}{K}_i-{\displaystyle \frac{1}{2}}10}^{J_f{\displaystyle \frac{1}{2}}{K}_f-{\displaystyle \frac{1}{2}}}\Delta =\sqrt{2}{C}_{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}10}^{{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}}=\sqrt{2}\sqrt{{\displaystyle \frac{1}{3}}}\Delta =\sqrt{{\displaystyle \frac{2}{3}}}\Delta \\ \left.〈J_f={\displaystyle \frac{1}{2}}{M}_f=-{\displaystyle \frac{1}{2}}{K}_f\right|{\widehat{D}}_{\mu =-1}\left|J_i={\displaystyle \frac{1}{2}}{M}_i={\displaystyle \frac{1}{2}}{K}_i\right.〉=\sqrt{{\displaystyle \frac{1}{3}}}\cdot \sqrt{{\displaystyle \frac{2}{3}}}\Delta ={\displaystyle \frac{\sqrt{2}}{3}}\Delta .\end{array}$$

(14)

In this case, we then have the operator matrices for the cyclic components of the rotating dipoles:

$$\begin{array}{l}{\widehat{D}}_{\mu =+1}=-\Delta {\displaystyle \frac{\sqrt{2}}{3}}\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)={\displaystyle \frac{2\Delta }{3}}\cdot {\displaystyle \frac{\left(-1\right)}{\sqrt{2}}}\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),\\ {\widehat{D}}_{\mu =0}={\displaystyle \frac{\Delta }{3}}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)={\displaystyle \frac{2\Delta }{3}}\cdot {\displaystyle \frac{1}{2}}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),\\ {\widehat{D}}_{\mu =-1}=\Delta {\displaystyle \frac{\sqrt{2}}{3}}\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)={\displaystyle \frac{2\Delta }{3}}\cdot {\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right).\end{array}$$

(15)

Then, for the Cartesian components we have:

$$\begin{array}{l}{\widehat{D}}_x={\displaystyle \frac{1}{\sqrt{2}}}\left({\widehat{D}}_{\mu =-1}-{\widehat{D}}_{\mu =+1}\right)={\displaystyle \frac{1}{\sqrt{2}}}{\displaystyle \frac{2\Delta }{3}}{\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)={\displaystyle \frac{\Delta }{3}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)={\displaystyle \frac{\Delta }{3}}{\sigma }_x,\\ {\widehat{D}}_y={\displaystyle \frac{i}{\sqrt{2}}}\left({\widehat{D}}_{\mu =-1}+{\widehat{D}}_{\mu =+1}\right)={\displaystyle \frac{i}{\sqrt{2}}}{\displaystyle \frac{2\Delta }{3}}{\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)={\displaystyle \frac{\Delta }{3}}\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)={\displaystyle \frac{\Delta }{3}}{\sigma }_y,\\ {\widehat{D}}_z={\displaystyle \frac{\Delta }{3}}\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)={\displaystyle \frac{\Delta }{3}}{\sigma }_z.\end{array}$$

(16)

Here σ_x,σ_y,σ_zare Pauli matrixes.

4.1. Hamiltonian and Eigentenergy of the System N Interacting Dipoles

The Hamiltonian for a system of N pairwise-interacting dipoles rotating in a crystal field is given by:

$$\begin{array}{l}\widehat{H}={\displaystyle {\displaystyle \sum _{i<j=1}^N}{\widehat{V}}_{\mathit{ij}}}={\displaystyle \frac{{D_0}^2}{4R^3}}{\displaystyle {\displaystyle \sum _{i<j=1}^N}\left(2{\widehat{j}}_i\cdot {\widehat{j}}_j-6{\widehat{j}}_{\mathit{in}}\cdot {\widehat{j}}_{\mathit{jn}}\right)}=\\ ={\displaystyle \frac{{D_0}^2}{4R^3}}\left\{\left({\widehat{J}}^2-{\displaystyle \sum _i^N{{\widehat{j}}_i}^2}\right)-3\left({\widehat{J}}_n^2-{\displaystyle \sum _i^N{{\widehat{j}}_{\mathit{in}}}^2}\right)\right\},\\ \widehat{J}={\displaystyle \sum _{i=1}^N{\widehat{j}}_i},\_\_\_{\widehat{J}}^2=\widehat{J}\cdot \widehat{J}.\end{array}$$

(37)

Here the operator of the quadrat of the total angular momentum of the system ${\widehat{J}}^2$ commutates with its projection on to quantization axe ${\widehat{J}}_n$: $\left[{\widehat{J}}^2{\widehat{J}}_n\right]=0$ and with each operator ${{\widehat{j}}_i}^2$, $\left[{\widehat{J}}^2{{\widehat{j}}_i}^2\right]=0$. We average the Hamiltonian (37) over the state $\left|J,{\widehat{J}}_n,j_1,j_2\dots j_N\right.〉$, then we have

$$\begin{array}{l}〈E〉={\displaystyle \frac{{D_0}^2}{4R^3}}\times \left\{\left(J\left(J+1\right)-N\cdot 1\cdot \left(1+1\right)-3M^2\right)\right.+\\ \left.+3\left.〈J,M,j_1=1\dots j_N=1\right|\left({\displaystyle \sum _i^N{{\widehat{j}}_{\mathit{in}}}^2}\right)\left|J,M,j_1=1\dots j_N=1\right.〉\right\}\end{array}$$

(38)

For most preferable state with the maximal value J = N ⋅ 1, the preferable projection M = N ⋅ 1 which realized the only way, when m_i = 1, then the energy of the ground state of the system with respect to the dipole-dipole interaction is

$$\begin{array}{l}{〈E〉}_0={\displaystyle \frac{{D_0}^2}{4R^3}}\left\{\left(N\left(N+1\right)-2N-3N^2\right)+3N\right\}=\\ ={\displaystyle \frac{{D_0}^2}{4R^3}}\left\{\left(N^2+N-2N-3N^2\right)+3N\right\}={\displaystyle \frac{{D_0}^2}{4R^3}}\left(-2N^2+2N\right)=\\ =-{\displaystyle \frac{{D_0}^2}{4R^3}}2N\left(N-1\right)\approx -{\displaystyle \frac{{\left(D_{0}N\right)}^2}{2R^3}}.\end{array}$$

(39)

This model demonstrates the possibility of forming a macroscopic dipole moment, (ND₀) within a domain. Such a polarized state is the most energetically favorable, specifically when accounting for dipole-dipole interactions.

4.2. Variation Model of Spontaneous Polarization in a Ferroelectric

We shall now demonstrate that, when finite temperature is taken into account, the formation of spontaneous dipole ordering indeed leads to a phase transition at a specific critical temperature. (All subsequent formulas will be written using temperature on an energy scale—that is, T = k_B T⁰, where k_B is the Boltzmann constant.)

Let N₊ be the number of rotating dipoles oriented upward, and N_- be the number of "inverted" dipoles oriented downward. Then (N₊ - N_-)D = Nd, where d is the average value of the angular momentum per dipole—that is,

$$\begin{array}{l}\underset{\_}{d}=D{\displaystyle \frac{N_{+}-N_{-}}{N}},\\ N_{\pm }={\displaystyle \frac{N}{2}}\left(1\pm {\displaystyle \frac{\underset{\_}{d}}{D}}\right).\end{array}$$

(40)

The resulting interaction energy of the entire system $〈E(\underset{\_}{d},T)〉$ can be rewritten as energy per dipole, as follows:

$${\epsilon }_{\mathrm{int}}(\underset{\_}{d},T)={\displaystyle \frac{〈E(\underset{\_}{d},T)〉}{N}}=-N{\displaystyle \frac{{\underset{\_}{d}}^2}{2R^3}}$$

(41)

To a given configuration of dipoles with a mean value d (40) and a mean energy per dipole defined by (41), there corresponds a partition function

$$\begin{array}{l}z={\displaystyle \frac{N!}{N_{+}!N_{-}!}}\exp \left(-{\displaystyle \raisebox{1ex}{${\epsilon }_{\mathrm{int}}$}\!\left/ \!\raisebox{-1ex}{$T$}\right.}\right)={\displaystyle \frac{N!}{N_{+}!N_{-}!}}\exp \left({\displaystyle \raisebox{1ex}{$N{\underset{\_}{d}}^2$}\!\left/ \!\raisebox{-1ex}{$2R^{3}T$}\right.}\right)\approx \\ \approx \sqrt{{\displaystyle \frac{N}{N_{+}{N}_{-}}}}{\left({\displaystyle \frac{N}{e}}\right)}^N{\left({\displaystyle \frac{e}{N_{+}}}\right)}^{N_{+}}{\left({\displaystyle \frac{e}{N_{-}}}\right)}^{N_{-}}\exp \left({\displaystyle \raisebox{1ex}{$N{\underset{\_}{d}}^2$}\!\left/ \!\raisebox{-1ex}{$2R^{3}T$}\right.}\right).\end{array}$$

(42)

(Here Sterling’s formula for factorial was used.) Then the free energy of this configuration system will be

$$\begin{array}{l}F=-\mathit{Tlnz}=-T\left\{{\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{N}{N_{+}{N}_{-}}}+N_{+}\ln {\displaystyle \frac{N}{N_{+}}}+N_{-}\ln {\displaystyle \frac{N}{N_{-}}}+\left({\displaystyle \raisebox{1ex}{$N{\underset{\_}{d}}^2$}\!\left/ \!\raisebox{-1ex}{$2R^{3}T$}\right.}\right)\right\}\approx \\ \approx -\mathit{NT}\left\{-{\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{\underset{\_}{d}}{D}}\right)\ln {\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{\underset{\_}{d}}{D}}\right)-{\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\underset{\_}{d}}{D}}\right)\ln {\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\underset{\_}{d}}{D}}\right)+{\displaystyle \raisebox{1ex}{${\underset{\_}{d}}^2$}\!\left/ \!\raisebox{-1ex}{$2R^{3}T$}\right.}\right\}=\\ =\mathit{NT}\left\{{\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{\underset{\_}{d}}{D}}\right)\ln {\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{\underset{\_}{d}}{D}}\right)+{\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\underset{\_}{d}}{D}}\right)\ln {\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\underset{\_}{d}}{D}}\right)-{\displaystyle \raisebox{1ex}{${\underset{\_}{d}}^2$}\!\left/ \!\raisebox{-1ex}{$2R^{3}T$}\right.}\right\}.\\ N_{\pm }={\displaystyle \frac{N}{2}}\left(1\pm {\displaystyle \frac{\underset{\_}{d}}{D}}\right).\end{array}$$

(43)

According to the extreme principle of thermodynamics, an equilibrium state corresponds to a minimum of free energy; that is, the derivative of the free energy with respect to any variable parameter must vanish upon the attainment of equilibrium. In this case, such a parameter is the average dipole moment per particle:

$$\begin{array}{l}{\displaystyle \frac{\partial F}{\partial \underset{\_}{d}}}=0=\mathit{NT}\left\{\ln {\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{\underset{\_}{d}}{D}}\right)-\ln {\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\underset{\_}{d}}{D}}\right)-{\displaystyle \raisebox{1ex}{$2\underset{\_}{d}D$}\!\left/ \!\raisebox{-1ex}{$R^{3}T$}\right.}\right\}=0,\\ \underset{\_}{d}=\mathit{Dth}\left({\displaystyle \raisebox{1ex}{$\underset{\_}{d}D$}\!\left/ \!\raisebox{-1ex}{$R^{3}T$}\right.}\right),\\ x=\left({\displaystyle \raisebox{1ex}{$\underset{\_}{d}D$}\!\left/ \!\raisebox{-1ex}{$R^{3}T$}\right.}\right),\\ {\displaystyle \frac{R^{3}T}{D^2}}x=\mathit{th}\left(x\right),\Rightarrow {\displaystyle \frac{R^{3}T}{D^2}}\le 1,\underset{\_}{d}\ne 0.\end{array}$$

(44)

From this, we obtain a transcendental equation for determining the equilibrium value $\underset{\_}{d}$:

$$\underset{\_}{d}=\mathit{Dth}\left({\displaystyle \raisebox{1ex}{$\underset{\_}{d}D$}\!\left/ \!\raisebox{-1ex}{$R^{3}T$}\right.}\right),$$

(45)

or, by introducing a new variable $x=\left({\displaystyle \raisebox{1ex}{$\underset{\_}{d}D$}\!\left/ \!\raisebox{-1ex}{$R^{3}T$}\right.}\right)$, we can rewrite the equation as

$${\displaystyle \frac{R^{3}T}{D^2}}x=\mathit{th}\left(x\right),$$

(46)

which yields a non-zero solution for x, generally for value d corresponding to spontaneous polarization $P=N\underset{\_}{d}$, provided that ${\displaystyle \frac{R^{3}T}{D^2}}\le 1$, i.e., at the critical temperature

$$T\le {\displaystyle \frac{D^2}{R^3}}=T*.$$

(47)

For the purpose of estimation, let us take the parameters of the unit crystal cell of barium hexaferrite, BaFe₁₂O₁₉, with dimensions a = 0.587 nm and c = 2.314 nm [9,10]. The characteristic temperature of the transition to the Para electric state (47) will be

$$T*~{\displaystyle \frac{{\left(2a_B\cdot e\right)}^2}{a^3}}={\displaystyle \frac{{\left(4,8\cdot {10}^{-10}\cdot {10}^{-8}\right)}^2}{{\left(5,87\cdot {10}^{-8}\right)}^3}}=1,139\cdot {10}^{-13}\mathit{erg}=813,57K$$

Evaluation of the critical temperature is in a good agreement with correspondent parameters of real materials (T~740K).

3. Results

Thus, we observe that a quantum model of interacting, rotating dipoles can account for the emergence of collective effects—such as ferroelectricity, characterized by established spontaneous polarization (where the magnitude of the polarization vector is proportional to the contribution of the dipoles involved in the collective excitation)—as well as the potential for the occurrence of an antiferroelectric effect. In both cases, electron spins play an important role; in the first instance, they are "free" and pair up between neighboring electrons, forming an antisymmetric state with a resultant spin of S=0, while the dipole moments align in parallel, which leads to an enhancement of electric polarization. On the other hand, in the case where the spin couples with the intrinsic rotation of a dipole to form a dipole-rotational state with angular momentum j=1/2, such dipoles—taking into account dipole-dipole (DD) interactions—pair up in an antiparallel fashion, thereby giving rise to a collective antiferroelectric effect (see equations (15)–(29)); the most energetically favorable state for two such dipoles is the state with total angular momentum J=0, M=0 (see (28)). The situation involving the recoupling of angular momenta is reminiscent of that which arises in diatomic molecules, in accordance with the various Hund’s coupling cases. Specifically, the latter scenario corresponds to Hund’s case (a), where the overall rotation of the molecule is sufficiently slow that the total molecular spin can adiabatically follow the molecular axis; in this case, the total spin interacts with an axially symmetric magnetic field directed along the molecular axis. In contrast, Hund’s case (b) occurs when the spin-orbit interaction approaches zero relative to the rotational motion—that is, when rotation dominates over fine-structure splitting. In this scenario, the spin of each dipole remains free, and the coupling of the two dipole spins is governed by the requirement that the total wavefunction of the two fermions be antisymmetric (see equations (30)–(36)). When dipole-dipole interaction is taken into account, the energetically most favorable state is the one with J=2, M=2 (see (32)). The statistical mean-field variational model is, in fact, developed specifically for this case, which features ferroelectric ordering.

4. Discussion

In ferroelectrics with an oxygen-octahedral structure, the ferroelectric transition is associated with the displacement of ions into new equilibrium positions that correspond to the structure of the ferroelectric phase. Research has shown that perovskite (BaTiO3) [10,11], pyrochlore (Cd2Nb2O7), and antimony sulfoiodide (SbSI) structures are conducive to a ferroelectric transition. In these crystals, the dielectric constant ε→∞ occurs at lower values of generalized polarizabilities than in crystals with a simple cubic structure. This result is consistent with experimental data: among crystals with these structures, there are indeed ferroelectrics. In the case of SbSI, the values of generalized polarizability for polarization along the С-axis (in an electric field E) are lower than those along the А and В axes. Consequently, in agreement with experimental data (at room temperature, ε_c = 50,000, while ε_a ~ε_b ~ 25), the ferroelectric transition occurs specifically along the C-axis. It has been experimentally confirmed that, in the vicinity of a ferroelectric transition, one of the transverse optical vibration frequencies approaches zero. The properties of this so-called "soft" vibration mode have recently been studied using infrared spectroscopy and slow-neutron scattering techniques. It can be assumed that the absence of oscillation frequency along the polarization axis, combined with the preservation of transverse modes in the plane perpendicular to the polarization direction, also points to the presence of rotating dipoles precessing along the polarization axis. This circumstance is indicated by the experimental results from studies [12,13].