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 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 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

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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 ∗ [37]. We extend for the first time this analysis to the class 퐴퐾푛푝 = (−1)푛−푝퐴퐾−푛 −푝 퐿 퐿 of tetragonal materials. The absence of magnetic 퐴퐾푛 2푝+1 = 0 퐿 (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 = 𝜏퐼 퐾퐿 퐿+푠 퐴 푛 −푝 퐴 푛푝 In a single crystal of the 퐷4ℎ phase, it is sufficient to { (−1) 퐾퐿 = 𝜏푇 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, 𝜏퐼 = +1 or −1 respectively rotation axis be parallel to 푧 and one of the 𝜎푣-type mirror for a symmetric or antisymmetric tensor under space planes be normal to 푥. In the tetragonal ferroelectric inversion, and 𝜏푇 = +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. 0 휅퐼퐴 sin 훽 sin 훼 Solving the corresponding set of internal and external 퐶4푣 ( 0 ) ( 휅퐼퐴 sin 훽 cos 훼 ) equations allows to determine the general form of the 퐴0 푐 tensor Σ as a linear combination of a reduced number of real parameters (Tables I—VIII). This form depends on the order and the type of the tensor and on the crystal 0 휅퐼퐴 sin 훽 sin 훼 point group. 퐶2푣 ( 0 ) ( 휅퐼퐵 sin 훽 cos 훼 ) 퐴0 푐 III. VECTORIAL RESPONSES

Illumination can quantitatively and qualitatively modify any physical quantity that is characteristic of the TABLE II. Polarization 푷 (type [-1,1)], for crystals equilibrium state of the material. For instance, in an with point groups 퐷4ℎ, 퐶4푣 and 퐶2푣. The components are illuminated stationary state, some vector quantities may expressed as functions of the spherical angles α, β of the appear: current density 풋 (photovoltaic effect), electric wave vector with respect to the crystal axes. The polarization 푷 (optical rectification) and magnetization 푴 phenomenological coefficients 푎, 퐴, 퐵… are arbitrary real (photomagnetic effect). 풋 and 푴 vanish in all the phases quantities. Small letters (푐) are used to present the non- when the light is turned off, while 푷 preexists in the dissipative terms whereas capital letters (퐴, 퐵) present ferroelectric phases 퐶4푣 and 퐶2푣. These three vectors the dissipative ones. Column 1: Point group of the phase. exhibit distinct behaviors when they are illuminated Column 2: Equilibrium value of 푷 without illumination. because they obey distinct transformation laws with Column 3: Additional stationary value of 푷 under respect to space (퐼) and time (푅) reversals. 풋 is anti- illumination. symmetric under both 퐼 and 푅 (type [-1,-1], 푷 is Crystal Spontaneous 푷 Illuminated 푷 symmetric for 푅 and antisymmetic for 퐼 (type [-1,1]), and point [-1,1] [-1,1] conversely for 푴 (type [1,-1]). For completeness, we will group also describe the behavior of a vector 푨, symmetric with 0 퐵 sin 훽 cos 훼 respect to 퐼 and 푅 (type [1,1]), which has however less 퐷4ℎ ( 0 ) ( −퐵 sin 훽 sin 훼 ) physical examples than 풋, 푴 and 푷. Let us notice that in a 0 0 magnetic insulator, 풋 can be replaced with the toroidal moment 푻, which describes a combination of spins arising spontaneously in specific anitiferromagnets and has 0 퐴 sin 훽 sin 훼 exactly the same symmetry properties as 풋. Accordingly, it 퐶4푣 ( 0 ) ( −퐴 sin 훽 cos 훼 ) formally exhibits the same physical behavior, so that all 퐴0 퐶 cos 훽 the qualitative conclusions we will get for 풋 in a conductor or a semi-conductor can be immediately translated for 푻 0 퐴 sin 훽 sin 훼 in the non-ordered phases of a magnetic insulator. 퐶 ( 0 ) ( −퐵 sin 훽 cos 훼 ) We can calculate the values of these four vectors as 2푣 퐴0 퐶 cos 훽 presented in section 2 (Eq. 1,2,3). The results are shown in Tab. 1—4 for the three phases of tungsten bronzes, where Wigner expansions are restricted to the first order in 퐿(퐿 ≤ 1). A. Optical rectification

Table II shows that in the three phases of the tungsten TABLE I. Type [1,1] vector (symmetric under both bronzes, the additional polarization induced by light space and time reversals) for crystals with point groups corresponds only to dissipative terms, so that they are 퐷4ℎ, 퐶4푣 and 퐶2푣. The components are expressed as negligible at low intensity. In the paraelectric (퐷4ℎ) functions of the spherical angles 훼, 훽 of the wave vector phase, the induced polarization is always in the 푥 − 푦 with respect to the crystal axes. The phenomenological plane (Fig. 2). The 푥 − 푦 plane projection of 푷 is parallel coefficients 푎, 퐴, 퐵… are arbitrary real quantities. Small to the wave vector, except in the orthorhombic phase letters (푐) are used to present the non-dissipative terms (Fig. 4). When the wave vector is parallel to 푧, the in- whereas capital letters (퐴, 퐵) present the dissipative plane component of 푷 vanishes, while it is maximum ones. Column 1: Point group of the phase. Column 2: along 푧 in the ferroelectric phases (퐶4푣 and 퐶2푣). When Equilibrium value of 푨 without illumination. Column 3: the wave vector is normal to 푧, the polarization also lies Additional stationary value of 푨 under illumination. within the 푥 − 푦 plane (Fig. 2—4). Crystal Spontaneous 푨 Illuminated 푨 point [1,1] [1,1] group 0 0 퐷4ℎ ( 0 ) ( 0 ) 0 푐

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FIG. 2. Induced polarization 푷, magnetization 푴 and electric current density 퐣 in the paraelectric phase 퐷4ℎ. (a) Wave vector 풌 parallel to +푧 . (b) 풌 in the 푥 − 푦 plane. (c) 풌 parallel to – 푧. FIG. 3. Optical rectification, photomagnetic and photovoltaic effects in the tetragonal ferroelectric (퐶4푣) phase under illumination. 풌 is the wave vector, 푷 the TABLE III. Magnetization 푴 (type [1,-1)], for crystals induced polarization, 푴 the induced magnetization and 풋 with point groups 퐷4ℎ, 퐶4푣 and 퐶2푣. The components are the induced electric current. Upper row: in the tetragonal expressed as functions of the spherical angles 훼, 훽 of the domain with up spontaneous polarization 푷ퟎ. Down row: wave vector with respect to the crystal axes. The in the tetragonal domain with down spontaneous phenomenological coefficients 푎, 퐴, 퐵… are arbitrary real polarization 푷ퟎ. (a) 풌 parallel to +푧, (b) 풌 normal to 푧. (c) quantities. Small letters (푐) are used to present the non- 풌 parallel to – 푧. dissipative terms whereas capital letters (퐴, 퐵) present the dissipative ones. Column 1: Point group of the phase. Column 2: Equilibrium value of 푴 without illumination. Column 3: Additional stationary value of 푴 under TABLE IV. Electric current 풋 and toroidal moment 푻 illumination. (type [-1,-1)], for crystals with point groups 퐷4ℎ, 퐶4푣 and Crystal Spontaneous 푴 Illuminated 푴 퐶2푣. The components are expressed as functions of the point [1,1] [1,1] spherical angles 훼, 훽 of the wave vector with respect to group the crystal axes. The phenomenological coefficients 푎, 퐴, 퐵… are arbitrary real quantities. Small letters (푐) are 0 0 used to present the non-dissipative terms whereas capital letters (퐴, 퐵) present the dissipative ones. Column 1: 퐷4ℎ ( 0 ) ( 0 ) 0 퐶 Point group of the phase. Column 2: Equilibrium value of 풋, 푻 without illumination. Column 3: Additional stationary value of 풋, 푻 under illumination. 0 휅퐼푎 sin 훽 sin 훼 Crystal Spontaneous 풋, 푻 [- Illuminated 풋, 푻 퐶4푣 ( 0 ) ( 휅퐼푎 sin 훽 cos 훼 ) point 1,-1] [-1,-1] 0 퐶 group

0 푏 sin 훽 cos 훼 0 휅 푎 sin 훽 sin 훼 퐼 퐷4ℎ ( 0 ) ( −푏 sin 훽 sin 훼 ) 퐶2푣 ( 0 ) ( 휅퐼푏 sin 훽 cos 훼 ) 0 0 0 퐶

0 푎 sin 훽 sin 훼 퐶4푣 ( 0 ) ( −푎 sin 훽 cos 훼 ) B. Photomagnetic effect 0 푐 cos 훽

The spontaneous magnetization vanishes in all the considered phases. The non-dissipative terms that appear 0 푎 sin 훽 sin 훼 ( ) under illumination in the ferroelectric phases are normal 퐶2푣 0 ( −푏 sin 훽 cos 훼 ) to the wave vector. They vanish when 풌 is parallel to 푧, 0 푐 cos 훽 and are maximum when 풌 is normal to 푧. Their direction is reversed when one considers two domains with opposite spontaneous polarizations (Fig. 2—4). C. Photovoltaic effect The 푧-component of 푴 is always dissipative and isotropic: it does not depend on the propagation direction The light-induced current exhibits only non- of the wave. Moreover, it is independent on the sense of dissipative terms independent of the ferroelectric the spontaneous polarization (Fig. 3). domains. The 푥 − 푦 components are normal to the wave

4 vector, except in the orthorhombic phase. They vanish letters (푎, 푓, … ) are used to present the non-dissipative when 풌 is parallel to 푧 (Fig. 2—4). terms whereas capital letters (퐷, 훥, … ) present the The 푧-component of 풋 vanishes in the non-polar phase dissipative ones. Roman and Greek letters represent the and when the wave is normal to 푧 in the polar phases. It is symmetric and skew-symmetric parts of the tensors maximum when the wave is parallel to 푧. In this case it respectively. changes its sign when the sense of propagation is Crystal reversed (Fig. 2—4). point Response tensors [1,1] group

푎 퐷4ℎ ( 푎 ) 푓

푎 0 (퐷 + Δ)휅퐼 sin 훽 cos 훼 ( 0 푎 −(퐷 + Δ)휅퐼 sin 훽 sin 훼) 퐶4푣 (퐷 − Δ)휅퐼 sin 훽 cos 훼 (−퐷 + Δ)휅퐼 sin 훽 sin 훼 푓

푎 + 퐴 cos 훽 휅퐼 0 (퐷 + Δ)휅퐼 sin 훽 cos 훼 ( 0 푐 + 퐶 cos 훽 휅퐼 (퐸 + Ε)휅퐼 sin 훽 sin 훼 ) 퐶2푣 (퐷 − Δ)휅퐼 sin 훽 cos 훼 (퐸 + Ε)휅퐼 sin 훽 sin 훼 푓 + 퐹 cos 훽 휅퐼

FIG. 4. Optical rectification, photomagnetic and photovoltaic effects in the orthorhombic ferroelectric TABLE VI. Matrices of the tensors of type [-1,1] (anti- (퐶2푣) phase under illumination. 풌 is the wave vector, 푷 symmetric under space reversal and symmetric under the induced polarization, 푴 the induced magnetization time-reversals) for crystals with point groups 퐷4ℎ, 퐶4푣 and 풋 the induced electric current. Upper row: In one and 퐶2푣. The response coefficients are expressed as orthorhombic domain with up spontaneous polarization functions of the spherical angles 훼, 훽 describing the 푷ퟎ. Down row: In the corresponding orthorhombic orientation of the wave vector with respect to the crystal domain with down spontaneous polarization 푷ퟎ. (a) 풌 axes. The phenomenological coefficients 푎, 푓, 퐷, … are parallel to +푧, (b) 풌 normal to 푧. (c) 풌 parallel to – 푧. arbitrary real quantities. Small letters (푎, 푓, … ) are used to present the non-dissipative terms whereas capital IV. TENSORIAL RESPONSES letters (퐷, 훥, … ) present the dissipative ones. Roman and Greek letters represent the symmetric and skew- 푖푗 Second-order tensors 퐾 (푖, 푗 = 푥, 푦, 푧), which play a symmetric parts of the tensors respectively. central role in material physics as response coefficients Crystal associated with various external excitations, may also be point Response tensors [-1,1] classified according to their parity with respect to space group and time reversals yielding four types of tensors. 푖푗 푗푖 0 Γ cos 훽 (퐷 + Δ) sin 훽 sin 훼 Moreover, symmetric ( 푆 = 푆) from skew-symmetric 퐷4ℎ 푖푗 푗푖 ( −Γ cos 훽 0 −(퐷 + Δ) sin 훽 cos 훼) ( 퐴 = − 퐴) tensors must be distinguished. Thus, one (퐷 − Δ) sin 훽 sin 훼 (퐷 − Δ) sin 훽 cos 훼 0 must evaluate the Wigner forms (Eq. 1) corresponding to eight types of such tensors. For instance, the dielectric constant 푖푗휀 is a (1,1) type symmetric tensor invariant 0 𝜍휅 + Γ cos 훽 (퐷 + Δ) sin 훽 sin 훼 under the two parities, whereas the magnetoelectric 퐼 퐶 ( −𝜍휅퐼 − Γ cos 훽 0 −(퐷 + Δ) sin 훽 cos 훼) 푖푗 4푣 response ( 훼) is a (-1,-1) type tensor antisymmetric (퐷 − Δ) sin 훽 sin 훼 (−퐷 + Δ) sin 훽 cos 훼 0 under time and space reversals containing both symmetric and skew-symmetric contributions. Solving the corresponding selections rules provides 0 푔휅퐼 + (퐵 + Γ) cos 훽 (퐷 + Δ )sin 훽 sin 훼 the Wigner expansions presented in Tables V—VIII up to (−푔휅퐼 + (퐵 − Γ) cos 훽 0 (−퐹 + E) sin 훽 cos 훼) 퐶2푣 first order in 퐿, where dissipative and non-dissipative (퐷 − Δ) sin 훽 sin 훼 −(퐹 + E) sin 훽 cos 훼 0 terms are distinguished.

TABLE V. Matrices of the tensors of type [1,1] TABLE VII. Matrices of tensors of type [1,-1] (anti- (symmetric under both space and time reversals) for symmetric under time reversal and symmetric under crystals with point groups 퐷4ℎ, 퐶4푣 and 퐶2푣. The response space inversion) for crystals with point groups 퐷4ℎ, 퐶4푣 coefficients are expressed as functions of the spherical and 퐶2푣. The response coefficients are expressed as angles 훼, 훽 describing the orientation of the wave vector functions of the spherical angles 훼, 훽 describing the with respect to the crystal axes. The phenomenological orientation of the wave vector with respect to the crystal coefficients 푎, 푓, 퐷, … are arbitrary real quantities. Small axes. The phenomenological coefficients 푎, 푓, 퐷, … are 5 arbitrary real quantities. Small letters (푎, 푓, … ) are used variety of phenomena that we will illustrate on to present the non-dissipative terms whereas capital considering only a few cases in the limit of low beam letters (퐷, 훥, … ) present the dissipative ones. Roman and intensity, when one may restrict the Wigner expansions Greek letters represent the symmetric and skew- to their dominant non-dissipative terms. One physical symmetric parts of the tensors respectively. example for each type of tensor will be discussed: strain Crystal tensor (type [1,1]), magnetoelectric tensor (type [-1,-1]), point Response tensors [1,-1] electric (type [1,-1]) and magnetic (type [-1,1]) group conductivities.

퐴 A. Photoelastic and photorefractive effects 퐷4ℎ ( 퐴 ) 퐹 The symmetry breakdown induced by illumination yields the onset of non-trivial components of the strain 푖푗 tensor 푇. Its non-dissipative terms can be deduced from 퐴 0 (푑 + 훿)휅퐼 sin 훽 cos 훼 Table V: ( 0 퐴 −(푑 + 훿)휅퐼 sin 훽 sin 훼) 퐶4푣 (푑 − 훿)휅퐼 sin 훽 cos 훼 (−푑 + 훿)휅퐼 sin 훽 sin 훼 퐹 푎 푚 푇 = ( 푎 ) + ( −푚 ) 푓 0 ′ 퐴 + 푎 cos 훽 휅퐼 0 (푑 + 훿)휅퐼 sin 훽 cos 훼 ′ ( 0 퐶 + 푐 cos 훽 휅퐼 (푒 + 휀)휅퐼 sin 훽 sin 훼 ) where the first matrix corresponds to the induced 퐶2푣 ′ (푑 − 훿)휅퐼 sin 훽 cos 훼 (푒 − 휀)휅퐼 sin 훽 sin 훼 퐹 + 푓 cos 훽 휅퐼 deformation in the 퐷 phase whereas the second matrix 4ℎ appears in the 퐶2푣 phase. The effect is isotropic in the sense that no dependence upon the wave direction is observed. Anisotropic effects appear nevertheless at high TABLE VIII. Matrices of the tensors of type [-1,-1] intensity (Table V) where we can no longer neglect (anti-symmetric under both space and time reversals) for dissipative terms. crystals with point groups 퐷4ℎ, 퐶4푣 and 퐶2푣. The response Thus, one sees that in the three phases the coefficients are expressed as functions of the spherical illumination effect is trivial, i.e. it does not break the angles 훼, 훽 describing the orientation of the wave vector symmetry (at this order of approximation) and cannot be with respect to the crystal axes. The phenomenological distinguished from thermal expansion. In addition, the coefficients 푎, 푓, 퐷, … are arbitrary real quantities. Small effect is exactly the same in the two types of ferroelectric letters (푎, 푓, … ) are used to present the non-dissipative domains in 퐶2푣 and 퐶4푣 phases. These conclusions also terms whereas capital letters (퐷, 훥, … ) present the hold for the dielectric and diamagnetic susceptibility dissipative ones. Roman and Greek letters represent the tensors because they are of the same type as the strain symmetric and skew-symmetric parts of the tensors tensor. However, one sees in Table VIII that non- respectively. dissipative terms have more complex effects in the two Crystal Response tensors [-1,-1] ferroelectric phases where shear strain is produced by point the light wave, and different effects occur in distinct group domains. The refractive index 푛 is also a type [1,1] second-order 퐷4ℎ 0 훾 cos 훽 (푑 + 훿) sin 훽 sin 훼 tensor and follows the same behavior under illumination ( −훾 cos 훽 0 (푑 + 훿) sin 훽 cos 훼) as the elastic deformation. Its light-induced modifications (푑 − 훿) sin 훽 cos 훼 (푑 − 훿) sin 훽 sin 훼 0 define the photorefractive effect, which provides a variety of non-linear optic phenomena. Thus, at the symmetry point of view, the refractive effect is qualitatively trivial 퐶4푣 퐴휅퐼 훾 cos 훽 (푑 + 훿) sin 훽 sin 훼 when one considers only non-dissipative terms. At larger ( ) ( −훾 cos 훽 퐴휅퐼 푑 + 훿 sin 훽 cos 훼) beam intensity dissipative terms become relevant and (푑 − 훿) sin 훽 cos 훼 (푑 − 훿) sin 훽 sin 훼 퐹휅 퐼 can provoke qualitatively non-trivial effects. The type [1,1] tensor symmetric contribution presented in Table V gives the form of the refractive 퐶 퐴휅 훾 cos 훽 (푑 + 훿) sin 훽 sin 훼 2푣 퐼 tensor in the 퐶4푣 phase that reads, taking into account ( −훾 cos 훽 퐷휅퐼 (푒 + 휀) sin 훽 cos 훼) zero-beam and dissipative digonal terms (푛푥, 푛푦, 푛푧), on (푑 − 훿) sin 훽 cos 훼 (푒 − 휀) sin 훽 sin 훼 퐹휅퐼 the one hand, and non-dissipative light-induced non- diagonal components, on the other hand:

푛 0 퐷휅 sin 훽 cos 훼 The phenomenological coefficients 푎, 푏, 퐷, Δ, … 푥 퐼 0 푛 퐷휅 sin 훽 sin 훼 appearing in these tensor matrices are let undetermined 푛 = ( 푦 퐼 ) by our symmetry analysis. They do not depend on the 퐷휅퐼 sin 훽 cos 훼 퐷휅퐼 sin 훽 sin 훼 푛푧 wave orientation but depend on the beam intensity and frequency and on the material properties and Since the effects of the diagonal terms have already temperature. These tensors allow to foresee a large been discussed above, let us focus on the symmetry- breaking non-diagonal terms. When the beam is not 6 parallel to 푧 they induce biaxiality together with a rotation of the principal optic axes. In the direction 풌’ normal to the projection of 풌 onto the optically isotropic 푥 − 푦 plane (Fig. 5a), the first principal refractive index is not modified by the illumination. The two other indices become (to the second order in the small parameter 퐷):

2 2 ′ 퐷 ′ 퐷 푛푥 = 푛푥 + , 푛푧 = 푛푧 − (4) 푛푥−푛푧 푛푥−푛푧

(assuming 푛푧 < 푛푥), which take the same values in the two domains of polarization (휅퐼 = ±1). On the other hand, the rotation of the two corresponding principal optic axes around 풌’ are opposite in these two domains (Fig. 5b). The rotation angle 훼 is given by:

퐷휅 tan 훼 = 퐼 (5) 푛푥−푛푧

Thus, even though the light wave propagates along 푥, 푦 or 푧 it is never parallel to any modified principal optic ′ axis (푧’, 풌⊥ and 풌’ in Fig. 5a) and splits into an ordinary FIG. 5. (a) Optic tensor 푛 without (left) and with and an extraordinary wave. Oppositely, an additional test (right) illumination. 풌 is the wave vector of the main ray with small intensity and traveling parallel to 풌’ is not beam. 풌⊥ is its projection onto the 푥 − 푦 plane, 풌’ the splitted. vector of the 푥 − 푦 plane normal to 풌⊥. The principal axes ′ Similarly, when the second ray is not parallel to 풌’, it of 푛 under illumination are 풌’ and two vectors 푧’ and 풌⊥ feels the induced biaxiality of the optic tensor and splits slightly rotated with respect to 푧 and 풌⊥. (b) 풌 is parallel into distinct rays. As an illustration of this non-linear to 풙 and the wave vector 풌ퟏ of the test beam parallel to 풚. optic effect, let us consider the main beam parallel to 푥 The principal axes of 푛 are 풚 and 푥’, 푧’ rotated in the 푥 − 푧 and the secondary beam along 푦 (Fig. 5b). The light- plane clockwise if the spontaneous polarization 푷ퟎ is up, modified principal optic axes are (i) along 푦, (ii) in the 푥 − and anticlockwise if 푷ퟎ is down. The test beam is not 퐷휅퐼 푧 plane, (iii) rotated by an angle 휃 = with respect to refracted. (c) 풌 is parallel to 푥 and the wave vector 풌ퟏ of 푛푥−푛푧 푥 and 푧 (assuming 퐷 small). Then, no splitting is observed the test beam parallel to z. The test beam is refracted into for the secondary wave, whereas when it is along 푧 (Fig. an ordinary wave along z and an extraordinary lying in 5c) it splits into an ordinary wave that remains parallel to the 푥 − 푧 plane wave deviated with respect to 푧 by an 푧 and an extraordinary wave that is deviated in one sense angle depending on the sense of 푷ퟎ. when the spontaneous polarization is along +푧, and to the opposite angle when the spontaneous polarization is B. Photomagnetoelectric effect along – 푧. Analogously, changing the sense of 풌 reverses the deviation angle of the extraordinary wave. Since the Under an applied magnetic field 푩 an additional induced biaxiality is a second-order effect in the small macroscopic polarization Δ푷 may be induced parameter 퐷 (Eq. 4) whereas the optic axis rotation is a proportionally to the field (and reciprocally a first-order effect (Eq. 5), one may neglect the former. The magnetization may appear under an applied electric refracted extraordinary wave lies then in the 푥 − 푧 plane field): Δ푷 = 휒푩. The corresponding magnetoelectric and its direction with respect to 푧 is inclined by an angle: tensor 휒 is of type [-1,-1]. In non-magnetically ordered phases the presence of time reversal in the macroscopic symmetry group of the crystal prevents the apparition of 휙푖푛푐 = 퐷휅퐼 (6) the magnetoelectric effect: 휒 = 0. Nevertheless, even in these materials the time reversal symmetry is broken by illumination, yielding non-zero magnetoelectric response, given in the studied phases by (see Table VIII):

0 훾 cos 훽 (푑 + 훿) sin 훽 sin 훼 ( −훾 cos 훽 0 (푑 + 훿) sin 훽 cos 훼) (푑 − 훿) sin 훽 cos 훼 (푑 − 훿) sin 훽 sin 훼 0 0 0 (푚 + 휇) sin 훽 sin 훼 + ( 0 0 −(푚 + 휇) sin 훽 cos 훼) (푚 − 휇) sin 훽 cos 훼 (−푚 + 휇) sin 훽 sin 훼 0

where the first matrix represents the tensor in the 퐷4ℎ and 퐶4푣 phases, whereas the second matrix provides the components onsetting in the 퐶2푣 phase. Thus, a magnetoelectric effect is present in all the phases under illumination, even at low intensity. 7

Only dissipative terms, distinguishing the two types of Similarly, when 푩 is normal to 푧, Δ푷 remains parallel to 푧 ferroelectric domains, allow to distinguish the 퐷4ℎ from but it no longer vanishes when 푩 is normal to 풌. the 퐶4푣 phase. One notices that in the three phases the illumination has important symmetry breaking effects. The absence of diagonal components shows that applying the magnetic field along a given direction yields no additional polarization along the same direction (however, dissipative terms can give rise to such a result). When the wave travels in parallel to the fourfold symmetry axis of the crystal (푧-direction), the magnetoelectric tensor becomes:

0 훾 0 FIG. 7. Variations of the polarization increment 훥푷 in (−훾 0 0) the 푥 − 푦 plane when the magnetic field is parallel to the 0 0 0 푧 crystal axis and the light wave vector turns in the 푥 − 푦 plane in the tetragonal phases (black ellipse) and in the Applying the magnetic field 푩 along +푥 yields the orthorhombic phase (gray circle). polarization Δ푷 along – 푦, while applying 푩 along +푦 yields Δ푷 along +푥, with the same amplitude for Δ푷 in C. Photoelectric conductivity both cases. No effect is present when applying 푩 parallel to 푧. The electric conductivity 𝜎 is a symmetric tensor of Conversely, when the wave travels perpendicularly to type [1,-1] that is modified by the light beam. Table IX 푧, the tensor becomes: shows the form of 𝜎 with and without light (neglecting 0 0 (푑 + 훿) sin 훼 non-dissipative terms in the latter case). ( 0 0 (푑 + 훿) cos 훼) (푑 − 훿) cos 훼 (푑 − 훿) sin 훼 0 0 0 (푚 + 휇) sin 훼 TABLE IX. Electric conductivity tensor without + ( 0 0 −(푚 + 휇) cos 훼) (second column) and with (third column) illumination in (푚 − 휇) cos 훼 (−푚 + 휇) sin 훼 0 the three phases. The illumination contributions contain only the non-dissipative terms.

Then, applying the field parallel to 푧 yields an Crystal Spontaneous Illuminated 𝜎 additional polarization in the 푥 − 푦 plane. More precisely, point 𝜎 [1,-1] [1,-1] in the tetragonal phases (first matrix) the polarization group modulus does not depend on the direction of the wave vector 풌 in the 푥 − 푦 plane. Furthermore, its direction is 𝜎푥 0 0 0 𝜎 normal to that of the wave (Fig. 6a). Reciprocally, 퐷4ℎ ( 푦 ) (0 0 0) applying the field normal to 푧 yields Δ푷 parallel to 푧, with 𝜎푧 0 0 0 its modulus vanishing if 푩 is normal to 풌 (Fig. 6b).

𝜎푥 0 0 cos 훼 푑 휅 sin 훽 ( 0 0 − sin 훼) ( 𝜎푦 ) 퐼 퐶4푣 cos 훼 − sin 훼 0 𝜎푧

푎′ cos 훽 0 푑 sin 훽 cos 훼 𝜎푥 ′ 휅퐼 ( 0 푐 cos 훽 푒 sin 훽 sin 훼 ) 𝜎 푑 sin 훽 cos 훼 푒 sin 훽 sin 훼 푓′ cos 훽 퐶2푣 ( 푦 ) 𝜎푧

In the non-polar phase 퐷4ℎ no contribution to the conductivity is produced by the beam within the non- dissipative approximation. In the ferroelectric phase the FIG. 6. (a) Polarization increment 훥푷 induced by a beam breaks the symmetry of the tensor, which changes light beam in the 푥 − 푦 crystal plane under applied of sign in two domains of opposite spontaneous magnetic field 푩 parallel to 푧 in the tetragonal phases. (b) polarizations. Thus, the light-induced current increment Polarization increment 훥푷 induced by a light beam in the Δ풋 reverses its sense in the two domains submitted to the 푥 − 푦 crystal plane under applied magnetic field 푩 normal same electric field (while the zero-beam contribution is to 푧 in the tetragonal phases. not changed).

In the 퐶4푣 phase, the tensor vanishes if the beam is The single difference occurring in the orthorhombic parallel to 푧, whereas it is maximal when the beam is in phase 퐶2푣 (second matrix) is that the fourfold symmetry the 푥 − 푦 plane. Moreover, when the beam is not parallel (푥 − 푦 exchange) is lost. Thus, when the field is parallel to to 푧, the effect does not depend qualitatively on the beam 푧 the additional polarization has a modulus that depends orientation: when the electric field 푬 is in the 푥 − 푦 plane, on its direction (Fig. 6), and it is no longer normal to 풌. 8

Δ풋 is along 푧 (Fig. 8a), and, reciprocally, when the field is (giving rise to higher-order effects), the electron along 푧, 횫풋 is in the 푥 − 푦 plane (Fig. 8b). In addition, in acceleration vanishes so that its velocity obeys the latter case Δ풋is parallel to the in-plane projection of the wave vector 풌, whereas in the former case Δ풋 푒𝜏 풋풑풉 풗 = 푩 × 풗 + vanishes when 푬 is normal to this projection. 푚 푛푒

where 𝜏 is the relaxation time [38]. Solving this equation together with Eq. 7 at low magnetic fields yields:

푛푒2휏 풋 = 푬 (8) 풑풉 푚 풊풏풕

that is, a Drude-type photocurrent (푛 is the electron density in the conduction band) and yields the following form of the tensor 𝜎푚:

푖푛푡 푖푛푡 0 −퐸푧 퐸푦 FIG. 8. Current increment 훥풋 induced by a light beam 푛푒3휏2 𝜎 = ( 퐸푖푛푡 0 −퐸푖푛푡) (9) 푚 푚2 푧 푥 with wave vector 풌 under applied electric field 푬 in the 푖푛푡 푖푛푡 tetragonal phases. (a) 푬 normal to 푧. (b) 푬 parallel to 푧. −퐸푦 퐸푥 0

In the orthorhombic phase, the situation is similar to 𝜎푚 depends on the symmetry of the material via the internal electric field components, and on the light wave that in 퐶4푣 when 풌 is normal to 푧. Conversely, when 풌 is parallel to 푧 applying 푬 parallel to 푥, 푦 or 푧 yields 풋 intensity and direction via the light-excited charge carrier parallel to 푬, as without light (except the reversion of Δ풋 density 푛. In a more realistic model, the crystal in opposite domains). For an arbitrary direction of 풌 the anisotropy transforms the relaxation time 𝜏 into a various contributions described here above appear symmetric tensor proportional to the zero-beam simultaneously. conductivity. Thus 𝜎푚 in Eq. 9 becomes the product of a 2 풊풏풕 symmetric (𝜏 ) with an antisymmetric (푬 ×) tensors, D. Photo-Hall effect and magnetoresistance so that it contains both symmetric and antisymmetric contributions. When in addition, one takes into account Magnetic effects are usually observed in metals the photoinduced parts of the internal electric and submitted to a DC electric voltage generating a primary magnetic fields and the exact form of 풋풑풉, 𝜎푚 takes a more (longitudinal) current, which can either be modified by intricate form that we will now analyze at the the applied magnetic field 푩 (magnetoresistance), or give phenomenological level. rise to a transverse electric field compensating the For clarity, let us analyze the effects of the symmetric transverse current (Hall effect). Such phenomena can also and antisymmetric contributions to 𝜎푚 separately. Table happen in the absence of applied voltage when the X shows the forms of the antisymmetric contributions to material is illuminated. The initial current 풋풑풉 submitted 𝜎푚 in the three phases. to the magnetic force, is then induced by the electrons photo-excited in the conduction band and then TABLE X. Antisymmetric contributions to the accelerated by the internal electric field 푬풊풏풕 of the ferroelectric materials (photovoltaic effect). This process photomagnetic conductivity tensor in the three phases. provides an electric current approximately directed along Crystal the internal electric field. However, an additional point 𝜎푚 mechanism participates to this effect: the electrons of the group material are directly accelerated by absorption () giving rise to a current roughly 0 Γ cos 훽 Δ sin 훽 sin 훼 −Γ cos 훽 0 −Δ sin 훽 cos 훼 directed along the light wavevector. In these two 퐷4ℎ ( ) processes, the anisotropy of the material actually deviates −Δ sin 훽 sin 훼 Δ sin 훽 cos 훼 0 the current direction with respect to 푬풊풏풕 and 풌. All these effects are considered within our formalism, by 0 𝜍휅퐼 + Γ cos 훽 Δ sin 훽 sin 훼 introducing a new conductivity tensor 𝜎푚: −𝜍휅 − Γ cos 훽 0 −Δ sin 훽 cos 훼 퐶4푣 ( 퐼 ) −Δ sin 훽 sin 훼 Δ sin 훽 cos 훼 0 풋 = 풋풑풉 + 𝜎푚푩 (7)

0 푔휅 + Γ cos 훽 Δ sin 훽 sin 훼 where 풋풑풉 is the photovoltaic current studied in section 퐼 퐶 (−푔휅 − Γ cos 훽 0 E sin 훽 cos 훼) 3.3 and the « photomagnetic conductivity » 𝜎푚 is a 2푣 퐼 symmetric type [-1,1] tensor. −Δ sin 훽 sin 훼 −E sin 훽 cos 훼 0 Let us use a simple academic model to illustrate Eq. 7. In an isotropic semiconductor in a stationary state where one neglects the photoelectric part of the induced current, the optical rectification and the photomagnetization 9

The only non-dissipative terms (proportional to 푔) to preserve the tetragonal symmetry and to apply our correspond to the photovoltaic/Drude-type effect analysis at the macroscopic level. 풊풏풕 presented in Eq. 9. Indeed, 푬 is parallel to 푧 in the When the wave propagates parallel to 푧 (Fig. 9a) 풋풑풉 is 푥푦 ferroelectric phases and thus yields the term 𝜎푚 = 푔 that also parallel to 푧 (see Table IV) and the photomagnetic provides an electric current that changes sense when 푬풊풏풕 conductivity reads: reverses. All the other terms are dissipative and therefore can be neglected at low beam intensities. It is interesting 0 𝜍휅퐼 + Γ 0 to note that these antisymmetric dissipative 𝜎푚 = (−𝜍휅퐼 − Γ 0 0) (10) contributions are the same in the ferroelectric tetragonal 0 0 0

퐶4푣 phase and in the paraelectric tetragonal 퐷4ℎ phase, and have the same sense in domains with opposite Thus, applying the magnetic field parallel to 푦 yields spontaneous polarizations. That indicates the essentially an additional current contribution parallel to 푥. Since the photoelectric character of these contributions, which barrel has a finite transversal size, its 푦 − 푧 walls become reinforces their smallness, since they have a relativistic electrically charged and thus create a transversal electric origin [16]. Reversing the sense of 풌 reverses the sign of field 푬풄풐풎풑 that compensates the 푥 current. Apart from the corresponding current contributions. the small relativistic Γ term (note that Eq. 10 holds for a When the beam travels parallel to 푧, the 푥 − 푧 and 푦 − wave travelling along +푧, for the opposite beam sense Γ 푧 components of 𝜎푚 vanish. Accordingly, no additional should be replaced with −Γ) and the absence of applied current parallel to 푧 is provoked by the applied magnetic DC voltage, this effect is exactly similar to the field. Conversely, when the beam travels normally to 푧, conventional Hall effect. 푥푦 When the wave propagates parallel to 푥, 풋 is parallel the contributions to 𝜎푚 that are independent on the 풑풉 domain (proportional to Γ) vanish. Thus, applying a to 푦 (see Table IV) and the photomagnetic conductivity magnetic field parallel to 푥 yields a current parallel to 푦, reads: wich follows the sense of the spontaneous polarization (and reciprocally for 푦). 0 𝜍휅퐼 0 Table XI shows the forms of the symmetric 𝜎푚 = (−𝜍휅퐼 0 −Δ + 퐷) (11) contributions to 𝜎푚 in the three phases. 0 Δ − 퐷 0

Applying the magnetic field parallelly to 푥 (Fig. 9b) yields a current parallel to 푦 and the conventional Hall TABLE XI. Symmetric contributions to the effect: the 푥 − 푧 surface charge creates an internal electric photomagnetic conductivity tensor in the three phases. field 푬풄풐풎풑 parallel to 푦 compensating both the Crystal photovoltaic and Hall currents. Applying 푩 parallelly to 푦 point 𝜎푚 (Fig. 9c) yields a current with a conventional Hall type group contribution parallel to 푥 and a non-conventional contribution parallel to 푧 that does not need to be 0 0 퐷 sin 훼 compensated. sin 훽 ( 0 0 −퐷 cos 훼) 퐷4ℎ At last, when the beam propagates in an arbitrary 퐷 sin 훼 −퐷 cos 훼 0 direction and the field is applied parallel to 푥 (or 푦), one finds conventional contributions parallel to the field (푥), 0 0 퐷 sin 훼 and unconventional contribution normal to it (푦) and 퐶4푣 sin 훽 ( 0 0 −퐷 cos 훼) accordingly, compensating internal electric fields (Fig. 퐷 sin 훼 −퐷 cos 훼 0 9d) in the 푥 − 푦 plane. The non-conventional effects are

dissipative and relativistic, so that their amplitudes must be very small. 0 퐵 cos 훽 퐷 sin 훽 sin 훼

퐶2푣 ( 퐵 cos 훽 0 −퐹 sin 훽 cos 훼) 퐷 sin 훽 sin 훼 −퐹 sin 훽 cos 훼 0

All the contributions are dissipative and independent on the ferroelectric domains, betraying their photoelectric character and their relative smallness. The absence of diagonal terms shows that applying the field along any of the principal axes 푥, 푦 or 푧 yields a current normal to it. On the other hand, applying the wave with 풌 parallel to 푧 cancels out the tensor in the two tetragonal phases. In these phases, applying the magnetic field parallel to 푧 produces a current parallel to 푧, whereas when the field is normal to 푧 and 풌 in the 푥 − 푦 plane yields a current parallel to 풌. Let us finally describe the photo-induced Hall effect in a rectangular barrel of the 퐶4푣 phase. This shape permits 10

퐴 푚푝 FIG. 9. Photo-Hall effect in a tetragonal barrel. The 퐾퐿 can be obtained by simple Clebsh-Gordan electric current can flow across the 푥 − 푦 walls. The photo combinatorial rules. This is no longer the case when the induced current in the 푥 − 푦 plane is compensated by an transition order parameter is spanned by waves lying in internal electric field 푬풄풐풎풑 due to the surface charges on the volume of the Brillouin zone, so that the cell volume is the 푦 − 푧 or 푥 − 푦 walls. 풋풑풉 is the photovoltaic current in changed at the transition. This phenomenon occurs for the absence of an applied magnetic field. (a) wave vector instance in PKN. Indeed, at the 퐷4ℎ → 퐶2푣 transition the beam 풌 parallel to 푥 and applied magnetic field 푩 parallel elementary cell is doubled with 푎표푟푡 = 푎푡푒푡 + 푏푡푒푡 and to y. (b) 퐤 parallel to 푥 and 푩 parallel to 푥. (c) 풌 parallel to 푏표푟푡 = 푎푡푒푡 − 푏푡푒푡, so that the orthorhombic cell is rotated 푥 and 푩 parallel to 푦. (d) 풌 in an arbitrary direction and 푩 by 45° with respect to the tetragonal cell (Fig. 10b). parallel to 푥. Consequently, the comparison between the two phases in Tables I—VIII has to be careful: the 푥, 푦 coordinates of the V. DISCUSSION AND CONCLUSION orthorhombic cell actually match with the 푥 + 푦, 푥 − 푦 coordinates of the tetragonal phase. We have calculated the responses to illumination corresponding to all types of vectors and second-order tensors in three phases with tetragonal/paraelectric, tetragonal/ferroelectric, and orthorhombic/ferroelectric symmetries. This works covers a vast domain of applications because many physical effects are accounted for by its results and because a large variety of materials are involved in these classes. Let us notice that even though any material with the right symmetry is concerned and will develop automatically the calculated tensors, their actual measurability depends on the type of material. For instance, the photovoltaic effect essentially concerns semiconductors with an adapted band gap, the magnetic effects involve materials containing magnetic ions in their crystalline cell or doped with such ions, and finally we expect a sufficient amplitude for measuring the photo-Hall effect only in metals. The governing parameter that allows to determine which material is susceptible to exhibit large photo-effects is its ability to transfer the incident photo energy to the electronic system. Along this line, efficient photovoltaic crystals are of course the main candidates. Within this family, the famous tetragonal FIG. 10. Cell relationships between the three phases with symmetry groups D4h, C4v and C2v. (a) In the primary ferroelectric perovskites BaTiO3 and PbTiO3 are well suited, at least for non-magnetic effects. ferroelectric-ferroelastic model. (b) In PKN. The phase transitions between the three studied Let us now explore the role of the in-plane anisotropy structures play an important role that we should briefly by comparing some photo-effects in tetragonal discuss. We can see in Tables I—VIII that each symmetry ferroelectrics (퐶 ) materials (PKN, GKN) with their breakdown (퐷 → 퐶 → 퐶 ) yields additional terms in 4푣 4ℎ 4푣 2푣 analogues in trigonal ferroelectric (퐶 ) materials the induced tensors, corresponding to successive onset or 3푣 (LiNbO3, BiFeO3). The comparison shows that the enrichment of the associated physical properties. Each photovoltaic effect has precisely the same form to the first new coefficient 퐴퐾푚푝 has a well-defined tensor character 퐿 order in 퐿 in the two classes of systems. However, we (which depends on the vector or tensor nature of Σ, but expect differences at higher orders since larger values of also on the value of 퐿) allowing to consider it as a 퐿 allow the crystal to better adapt its responses to the secondary (non-symmetry-breaking) order parameter (at details of its anisotropy. Conversely, considering second- the center of the Brillouin zone) coupled to the primary order tensors, such as the photomagnetoelectric tensor, order parameter describing the transition [39]. The type shows strong differences between the tetragonal and of coupling between these two order parameters controls trigonal systems. 퐴 푚푝 the temperature dependence of the onsetting 퐾퐿 below In conclusion, we have systematically predicted the the transition. This coupling has not a universal feature form of the angular variations of four vectorial and four since the macroscopic symmetry breakdown sequence tensorial responses of tetragonal and orthorhombic 퐷4ℎ → 퐶4푣 → 퐶2푣 can be associated with a number of ferroelectric materials to linearly polarized light beam on associated space group sequences. using a new phenomenological method that cures some The simplest case arises when the first transition drawbacks of the traditional approach. We have applied (퐷4ℎ → 퐶4푣) is a primary ferroelectric transition (i.e., the our results to the phase sequence observed in the macroscopic polarization is the primary order polymorphism of tetragonal tungsten bronzes. We parameter) and the second transition (퐶4푣 → 퐶2푣) is discuss in detail some geometric aspects of the optical primary ferroelastic (Fig. 10a), which are realized in the rectification, the photomagnetism, the photovoltaic effect, GKN family. Since these order parameters lie at the center the photoelasticity, the photorefractive effect, the of the Brillouin zone, their coupling with the coefficients photomagnetoelectricity, the photoconductivity and the 11 photo-Hall effect. We have paid special attention to two tensor that, in turn, refracts the extraordinary ray of a unconventional effects: (i) The photo-Hall effect, in which secondary beam. Both effects are precisely described by the magnetic field deviates photo-excited electrons our approach with a small number of phenomenological instead of conducting electrons as in the conventional parameters that can be easily determined on probing Hall effect, and (ii) a non-linear optic effect in which a these effects. primary light beam controls the rotation of the optic

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