Photo-Hall, photorefractive and photomagnetoelectric effects in tungsten bronzes and related tetragonal ferroelectrics I. Tekaya1, A. Tekaya1, B. Maximin2 1 Laboratoire de Physique de la Matière Condensée, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex, France 2 Physique des Systèmes Complexes, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens Cedex, France We present an extensive study of the electric, magnetic and elastic responses of tetragonal ferroelectrics under illumination, using a new theory intertwining material and wave symmetries. Optical rectification, photomagnetic, photovoltaic and phototoroidal vector responses are worked out as functions of the wave vector and wave polarization directions. Second-order response tensors associated with photoelastic, photomagnetoelectric, photorefractive effects and photoconductivity are described. We discuss in detail the photo-Hall and a new non-linear optical effect in tungsten bronzes. Finally, we compare the properties of tetragonal materials with those of hexagonal, orthorhombic and trigonal ferroelectrics previously reported in the literature. I. INTRODUCTION former permit to calculate the polarization, magnetization, toroidal moment and electric currents The family of tetragonal tungsten bronze (TTB) induced (or modified for the polarization in ferroelectric materials, evidenced since 1949 and intensively studied phases) under illumination. The latter predict the light- both as single crystals and as ceramics [1—7], exhibits induced elastic deformation and the modifications of exceptional ferroelectric behaviors [8,9], together with various response coefficients: optic tensor, magneto- non-linear optical properties. These features make them electric tensor, conductivity and a Hall-type tensor. We interesting materials for electro-optics and pay special attention to the distinction between the optoelectronics [1,10—16] as well as promising effects due to dissipative and non-dissipative processes candidates for computer memory applications based on [19,31], respectively, which exhibit distinct symmetry resistive switching phenomena [17,18]. An additional properties. This theoretical approach has a advantage of these materials is the large variety of atoms phenomenological character and it is only based on the present in their cell, which permit many chemical symmetry of the light beam and that of the crystal. Thus, substitutions giving rise, for instance, to the possibility to our results have a wide range of applications, since they give magnetic properties to these structures. Thus, apply to any material with tetragonal (more precisely bronze tungstens potentially become multiferroic if they with point group 퐶4푣 or 퐷4ℎ) and possibly orthorhombic can undergo a magnetic ordering transition and can (퐶2푣) symmetries in their phase diagrams. acquire magnetic properties even in their non-magnetic We use two TTBs, namely the lead potassium niobate phases. Moreover, their stability under intense laser light Pb2KNb5O15 (PKN) and the gadolinium potassium niobate and their high non-linear coefficients open the possibility GdK2Nb5O15 (GKN), as typical examples of such materials. both for using illumination to efficiently probe these PKN is orthorhombic ferroelectric at room temperature materials, but also for exploiting them in order to [3]. Its space group is Cm2m and it undergoes a phase evidence new phenomena resulting from their coupling transition at 푇퐶 = 723 퐾, becoming paraelectric with with high-intensity light beams. P4/mbm symmetry. Its high Curie temperature proves We investigate theoretically such phenomena on very useful for optical applications such as waveguides applying an approach recently proposed [19] to predict and birefringence [32]. GKN, also ferroelectric at room the tensorial response of single crystals to linearly temperature with space group P4bm [5], undergoes two polarized light waves. We will focus attention only on the phase transitions at 511 K and 648 K related to the physical effects involving four types of vectors (section 3) ferroelectric-antiferroelectric-paraelectric phases [33]. and the four types of second-rank tensors (section 4), Indeed, its tetragonal space group in the ferroelectric symmetric or antisymmetric with respect to time and phase is P4bm and becomes P4nc in the antiferroelectric space reversals. phase, then switching to P4/mbm in the paraelectric We illustrate our results with the help of a limited phase [5]. The full polymorphism of PKN and GKN can be number of examples: optical rectification [20], summarized with only three point groups: the tetragonal photomagnetic [21], phototoroidal [22,23] and 4/mmm (퐷4ℎ), 4mm (퐶4푣) groups and the orthorhombic photovoltaic effects for vector responses, on the one 2mm (퐶2푣) group. The global structure of TTBs is hand, and photoelastic [24], photorefractive [25—27], presented on figure 1. photomagnetoelectric, photoconductivity [28] and photo- Hall effects [29,30] for tensors, on the other hand. The 1 We then apply the light beam to each domain and probe the modifications of any tensor Σ induced by the light beam. The stationary values of this tensor depend then on 휅퐼, on the one hand, and to the orientation of the light beam on the other hand. This orientation is characterized by the three Euler angles 훼, 훽, 훾 that permit to rotate the laboratory frame 푥, 푦, 푧 into the (normalized) wave frame 풆, 풃, 풌, where 풆 and 풃 are unit vectors proportional to the electric and magnetic polarizations of the linearly polarized wave, and 풌 is proportional to its wave vector. The cartesian components of the response tensor are thus functions 퐴 Σ (휅퐼, 훼, 훽, 훾) of the previous parameters. They can then 퐿 be expanded into Wigner spherical functions 퐷푚푝(훼, 훽, 훾): Σ퐴(휅, 훼, 훽, 훾) = +∞ 퐿 퐴 푚푝 퐴 푚푝 퐿 ∑퐿=0 ∑푚,푝=퐿{ 퐾퐿 (0) + 휅퐼 퐾퐿 (2)} 퐷푚푝(훼, 훽, 훾) (1) where the 퐴퐾푚푝(푠 = 0,2) are complex phenomenological 퐿 FIG. 1. Schematic projection of the structure of coefficients. 퐴 is a tensorial multi-index (퐴 = 푖 = 푥, 푦, 푧 tetragonal tungsten bronzes on the a—b plane, showing for vector responses and 퐴 = (푖, 푗) for second-order the pentagonal sites (A), the square sites (B) and the tensors), 퐿 = 0,1,2, … and – 퐿 ≤ 푚, 푝 ≤ 퐿 are integer triangular sites (C). indices similar to the 퐿 and 푚 indices of the spherical 푚 harmonics 푌퐿 . Therefore, expansion (1) appears to be We expose the theoretical formalism in section 2. analogous to a usual expansion of a function depending 푚 Then we apply this formalism to vectors and second- on the two spherical angles 휃, 휑 into 푌퐿 . Accordingly, order tensors in sections 3 and 4. Finally, we discuss our increasing 퐿 amounts to improve the angular precision of results and compare them with other materials in section Eq. 1. We will use in the sequel a simple approximation 5. that consists in restricting expansion (1) to 퐿 = 0 and 1, which provides the main angular variations of 퐾. The II. WIGNER EXPANSION same restriction was used in [36] for KBiFe2O5 and in [37] for BiFeO3, whereas a more accurate second-order The theoretical approach we use for studying approximation (퐿 = 0,1,2) was used in [19] for LiNbO3. illumination effects has been recently developed [19] in Even with small values of 퐿 the number of 퐴 푚푝 order to determine the light-induced modifications of independent coefficients 퐾퐿 (푠) is large. It can be physical tensors as functions of the wave vector and wave drastically reduced when considering the symmetry polarization orientations. Its formalism, based on group of the wave together with that of the crystal. Each expansions of the response coefficients in Wigner generator of these two groups acts linearly on the 퐴 푚푝 functions [34], was previously developed [35] within the coefficients 퐾퐿 (푠) and thus provides a linear equation framework of nematic liquid crystals, and was then for these parameters (the so-called external (wave) and applied to multiferroic materials such as the ferroelectric internal (crystal) selection rules of [19]). The external trigonal LiNbO3 [19], the photovoltaic orthorhombic rules read: KBiFe2O5 [36] and the rhombohedral monoclinic BiFeO3 퐴 푛푝 ∗ 푛−푝퐴 −푛 −푝 [37]. We extend for the first time this analysis to the class 퐾 = (−1) 퐾 퐿 퐿 of tetragonal materials. The absence of magnetic 퐴 푛 2푝+1 퐾퐿 = 0 (2) structures in their phase diagrams allowed us to slightly 퐿+푝 퐴 푛 −푝 퐴 푛푝 simplify the formalism of [19]. (−1) 퐾퐿 휑푠 = 퐼 퐾퐿 퐿+푠 퐴 푛 −푝 퐴 푛푝 In a single crystal of the 퐷4ℎ phase, it is sufficient to { (−1) 퐾퐿 = 푇 퐾퐿 consider a single domain. We orient this domain in the laboratory frame (푥, 푦, 푧) in such a way that the fourfold where 휑0 = 1 and 휑2 = −1, 퐼 = +1 or −1 respectively rotation axis be parallel to 푧 and one of the 푣-type mirror for a symmetric or antisymmetric tensor under space planes be normal to 푥. In the tetragonal ferroelectric inversion, and 푇 = +1 or −1 in the same manner for phase (퐶4푣), one has to consider simultaneously two time reversal. The fourth equation is related to time domains, since they respond differently to the light beam. reversal symmetry and permits to distinguish dissipative In one domain (domain 1) we proceed as in the 퐷4ℎ from non-dissipative terms in Eq. 1 [19,20]. phase. The other domain (domain 2) is defined by the The internal rule reads, for rotation symmetries: application of space inversion 퐼 to the first domain. We 퐴 +퐿 퐵 푚푛 퐿 −1 퐴 푛푝 associate an integer number 휅퐼 to each domain, where 푅퐵 ∑푚=−퐿 퐾퐿 (푠)퐷푚푝(푅 ) = 퐾퐿 (푠) (3) 휅퐼 = +1 for the domain 1, and 휅퐼 = −1 to the domain 2. 퐴 푖 Of course, the spontaneous polarization is reversed where 푅퐵 = 푅푗 is the rotation matrix in cartesian between the two domains. 푖 푘 coordinates (for vectors) or 푅푗 푅푙 (for tensors), whereas 2 space inversion transforms 휅퐼 into −휅퐼 and time reversal transforms 휅푇 into −휅푇.