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Magnetic linear birefringence of light in Tb3Ga5O12
Optical Materials 88 (2019) 103–110
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Magnetic linear birefringence of light in Tb3Ga5O12 a
b
c
T b
Rakhim Yu Rakhimov , Uygun V. Valiev , Denis N. Karimov , Ramil R. Vildanov , Dejun Fu
a,∗
a Key Laboratory of Artificial Micro- and Nano-Structures of the Ministry of Education of China and Hubei Key Laboratory of Nuclear Solid Physics at the School of Physics and Technology, Wuhan University, Wuhan, 430072, China b Faculty of Physics, National University of Uzbekistan, Tashkent, 100174, Uzbekistan c Federal Scientific Research Centre «Crystallography and Photonics», Russian Academy of Sciences, Moscow, 119333, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Magnetic linear birefringence Terbium-gallium garnet Akulov's relation
The spectra of the magnetic linear birefringence (MLB) in the terbium-gallium garnet TbGG (Tb3Ga5O12) have been studied within the visible spectral range for temperature T = 90 K. Analysis of the temperature and spectral dependences of MLB has made it possible to determine the “effective” frequencies of the allowed f→ d electric dipolar transitions in rare-earth (RE) ions of Tb3+ ions responsible for the formation of quadratic magnetooptical effect in this garnet. The results of the performed magneto-optical experiments in TbGG have confirmed the Akulov's relations (for the quadratic magnetic effects) arising from the phenomenological consideration of MLB for the values of Δn measured at the orientation of the external field in the plane (110) along the main crystallographic directions of the garnet crystal. From analysis of the experimental data on MLB was obtained that for the TbGG MLB changes quadratically with the field within the temperature range 90–300 K. At the same time with a decrease in temperature, its value grows inversely proportional to the square of temperature. A certain difference was also found in the “effective" frequencies of the allowed transitions in Tb3+ ions of the garnet, which is determined from the optical absorption spectra of TbGG on the one hand, and from the analysis of the spectral dependence of MLD on the other one. It has been assumed that this difference can be due to significant distortions of the TbGG crystal lattice caused by this significant magnetostriction at low temperatures, leading to the "mixing" of the 9D term states with the "optically resolved" states of the 7D term.
1. Introduction The magnetic linear birefringence (MLB) observed in the rare-earth ferrite-garnets (REFG) [1–3] and in a number of other crystals with different magnetic structures [4,5] is quite large, Δn ∼10−5–10−3. An investigation of this phenomenon in magnetically ordered crystals is of great interest since it gives a unique possibility of associating the optical characteristics of crystals with magnetic ones. Establishment of such a relation is important for studying the spin dependence of the crystal polarizability. Such a relation can be used intensively for the study, by optical methods, of the magnetic structure of crystals, magnetization of sublattices, magnetic phase transitions and other phenomena. At the same time, in the late 20th century, significant interest has been attracted to the MLB research in the paramagnetic rare-earth (RE) garnets [6,7]. By the numerous experimental studies carried out with the REFG [1–4], it was found that MLD in them is largely determined by a contribution of the subsystem of RE-ions and varies greatly in the transition from one ion to another. In particularly, in the REFG series, MLB can vary with temperature
∗
by various ways (up to the change of a sign of the effect). The temperature dependences of the MLB for different crystallographic directions, and even in the same garnet, can differ significantly from each other (see, for example [3,4]). The phenomenological theory operating with the quadratic dependence of MLB on magnetization experiences significant difficulties in the interpretation of the temperature, field and orientation dependencies of MLB in REFG [2–4]. Indeed, for their detailed analysis within the microscopic approach, it is necessary to know the wavefunctions and energy spectra of the rare-earth ion in garnet [6–8]: those parameters are now insufficiently studied. A complex magnetic structure of REFG is formed by a superposition of several magnetic sublattices each of which contributes to the MLD. In turn, the temperature-dependent contribution of a single sublattice is determined by its orientation relative to the crystal axes and direction of the external magnetic field, which greatly complicates the consideration of its contribution to MLD. Therefore, the reason for the difference in the temperature dependences of MLB in the REFG series is still not sufficiently clear. At the same time, in the systems simpler than the REFG, i.e. the rare-earth paramagnetic garnets (gallates and aluminates), a
Corresponding author. E-mail addresses: [email protected], [email protected] (D. Fu).
https://doi.org/10.1016/j.optmat.2018.11.022 Received 22 July 2018; Received in revised form 8 November 2018; Accepted 15 November 2018 0925-3467/ © 2018 Published by Elsevier B.V.
Optical Materials 88 (2019) 103–110
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Passing through the Faraday cell (FC) and analyzer (A), the light was focused on the input slit of a high resolution double diffraction monochromator (MDR model 23, LOMO, Russia). After that the passing light is incident on a cathode of the photomultiplier (PMT). The monochromator (M) have a spectral resolution generally better than 0.05 nm over the wavelength range under study. The state of light polarization was modulated with a frequency of Ω = 80 Hz by the Faraday cell [4,6] made from a terbium glass bar (10 cm in length). The signal was registered by a narrow-band (selective) amplifier (SA) and synchronous detector (SD). Magnetic birefringence occurs in a crystal as a result of the fact that → → → → → two light waves polarized with E // H and E ⊥ H , where E is the electric → vector of the light wave and H is the external magnetic field, propagating in the crystal perpendicular to the magnetization have different phase velocities. Plane-polarized light passed through the crystal has the electric vector oriented at an angle of 45° to the direction of the external magnetic field during the magnetic birefringence measure→ ments. The E components parallel and perpendicular to the external magnetic field are then equal in amplitude and the birefringence is only associated with the difference in the phase velocities of these components. The relationship between the phase difference β and the difference between the refractive indices Δn = n// − n⊥ is given by formula [4]:
similar problem has been resolved in the framework of microscopic representations developed in Refs. [6–8]. Indeed, according to [6–8], in the case of rare-earth magnetics, the MLB is determined by a RE-ion quadrupole momentum Q induced by an external magnetic (or effective exchange) field. In this case, the features of the MLB behavior are determined both by the electronic structure of the magnetoactive ions formed in the low-symmetry (symmetry D2) crystal environment of the RE-ion and by the changes in the quadrupole moment of the RE-ion. These changes in the Q arise from the Van Vleck's “mixing” [9] of the wavefunctions of the sublevels of the ground multiplet by the external magnetic field H. In this case, the MLB behavior also depends strongly on a character of the crystal field (CF) splitting of the Stark structure of the ground multiplet of the RE-ion in the garnet crystal. As shown in Ref. [6], the splitting of RE-ion multiplets in the crystal field of garnet-gallate (or aluminate) leads to the appearance of doublet levels for the so-called Kramers ions (Dy3+, Er3+, Yb3+) or singlet levels for non-Kramers ions (Eu3+, Tb3+, Tm3+, Ho3+), according to Kramers theorem [9,10]. This important circumstance is associated with the emergence of field and the temperature dependences of MLD are significantly different from similar dependences predicted in the framework of the traditional theory of quadratic magneto-optical phenomena [2,4,5]. In addition, an investigation of the MLB spectral dependence for REgarnets can also serve as an additional criterion for the adequacy of the theoretical constructions developed in Refs. [6,7]. The latter circumstance is very relevant for the terbium-gallium garnet Tb3Ga5O12 (TbGG) since the theoretical analysis of the field, spectral and temperature dependences of MLB in TbGG, measured in present work, can be substantially simplified in the limiting case of sufficiently high temperatures (≥100 K). The reason for this simplification is that, according to magnetic measurements [11], the magnetization of TbGG RE-garnet can be very well described by the approximation of a “free RE-ion” within the temperature range of 78–300 K. Therefore, a special character of the behavior of the TbGG magnetic properties [11] allowed us to interpret the data of MLB measurements in TbGG within the temperature range 90–300 K by the so-called approximation of a “free Tb3+-ion".
β=
2π Δn⋅l λ
(1)
here λ is the light wavelength in vacuum, l is the crystal length. If one of the axes of the Senarmont compensator is parallel to the transmission plane of the polarizer (P) then the angle θ between the transmission planes of the polarizer (P) and the analyzer (A) is close to 90 deg, i.e. θ = 900 ± Δ. Then the main component I (Ω) of the intensity of the light I incident on the photomultiplier photocathode will be as follows:
I (Ω) =
I0 [2J1 (2α 0 )⋅sin(2θ + β )⋅sin Ωt ] 2
(2)
where I0 is the light intensity at the analyzer; J1 (x ) is the first-order Bessel function [12,13]; α 0 = 1 − .1. 50 is the amplitude of rotation angle provided by the Faraday cell; Ω is the modulation frequency. It should be noted that in this method of measurement so-called “zeromethod” of MLB measurement can be realized. In this case, the signal I (Ω) induced by a magnetized sample can be compensated by the rotation of an output analyzer placed after the Senarmont compensator at an angle equal to Δ. As a result, the rotation angle of the analyzer is equal to half of the phase difference β caused by MLB, according to the β relation Δ = ± 2 . Single crystal of terbium-gallium garnet TbGG was grown by a method of spontaneous crystallization from a solution-melt (Czochralski method) and provided by Prof. B.V. Mill from Moscow State University (Russia). A sample was oriented by the XRD method and cut in the
2. Measurement technique and samples An experimental setup for the measuring of the magnetic linear birefringence spectra within the visible region was used. A 100W halogen lamp was utilized as a light source (see Fig. 1). The light from the halogen lamp (S) is collected by a quartz condenser (L) and goes through a polarizer (P) to a crystal (TbGG) mounted in a cryostat with controllable temperature. The cryostat is placed in a gap of the electromagnet (EM) with the magnetic field (H = 6.5 kOe) perpendicular to the direction of light propagation. The phase difference that the light acquired in the crystal was measured with an optical Senarmont compensator (λ/4) placed between a crossed polarizer and analyzer.
Fig. 1. S is the light source, L is the quartz lens, P is the polarizer, EM is the electromagnet, PS is the power supply for EM, TbGG is the garnet sample, λ/4 is the Senarmont compensator, FC is the Faraday cell, PA is the power amplifier for FC, G is the generator, A is the analyzer, PMT is the photomultiplier, A is the preliminary amplifier (with high resistance), SA is the selective amplifier, SD is the synchronous detector.
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crystals doped with non-Kramers ions is “mixing” of the wavefunctions of the closely located non-degenerate Stark singlets in the “quasidoublets” by an external field H. Indeed, from an analysis of the transverse and longitudinal Zeeman effect observed in absorption of the 5 D4 multiplet the authors of [20,21] identified the ground state of the Tb3+ ion in TbGG as a “quasidoublet” consisting of two lowest nondegenerate Stark levels. According to the CF calculations [21], the wavefunctions of this “quasi-doublet” can be described by “pure”/ J,m > states characterized by the maximum values of the angular momentum projections m = ± 6 and making the largest contribution to the average magnetic moment of the PE subsystem. Therefore, it is natural to expect that in the formation mechanism of the temperature dependence of both magnetization and MLD in terbium-gallium garnet, this quantum state will plays a main role. Since the presented-here studies consider not only the temperature and spectral dependences of MLB in TbGG, but also orientation ones, it seems appropriate to make a few detailed comments concerning the features of the crystallographic structure of RE-garnets. Terbium-gallium garnet is crystallographically cubic and belongs to the space group Ia3d. There are eight formula units per cell. The unit cell of terbiumgallium garnet crystal, containing 160 atoms, is shown in Fig. 2. The cations (Tb3+ and Ga3+) are surrounded by oxygen atoms in the various symmetry in the garnet unit cell and crystal structure of this compound can be represented in the coordination polyhedron form of the interconnected dodecahedrons, octahedrons and tetrahedrons with common oxygen atoms. The Tb3+ ions occupy (24c) positions and each Tb3+ ion is coordinated with eight oxygen ions in the so-called dodecahedral site (distorted cube form). The Ga3+ ions occupy the (16a) positions in octahedrons and tetrahedral (24d) positions in the crystal lattice. The oxygen ions O2− occupy the (96h) positions and each oxygen ion is a part of two dodecahedrons, one octahedron and one tetrahedron. The (110) crystal plane and positions of the specific crystallographic [001], [110] and [111] directions are schematically shown in Fig. 2. These specified directions are of interest for orientation-dependent magneto-optical investigation in the terbium-gallium garnet. From Fig. 2 it is also clearly seen that the crystallographic axes of the type [001] and [110], as well as the axes [110] and [111], are mutually perpendicular to each other, respectively. But then it can be expected that in the MLB the sum of changes in the refractive indices for the axes [001], [111] and [110] of the garnet cubic crystal will be equal to zero [6]. This statement can follows from the law of Akulov for the even (i.e. quadratic) magnetic effects as this law states that “the summation of the results of measurements of an even magnetic effects for the three arbitrary mutually perpendicular directions is equal to zero [22]". Indeed, for a number of the cubic crystals of rare-earth garnet, i.e. gallates RGG (where R = Eu, Tb, Er, Tm) and aluminates RAG (R = Dy, Ho, Er, Yb), the following relations for MLB at the different → → orientations of vectors H and k were found in Ref. [6] and experimentally confirmed in Ref. [23]:
crystallographic plane [110], its surface was polished by diamond pastes using slowly diminishing grain size (down to ∼ 1 mkm). The thickness of the TbGG sample used in the magneto-optical measurements was ∼1.5 mm. 3. Some features of the theoretical description of MLB in TbGG As shown earlier in Refs. [6–8], for the phase shift β arising from the MLB to be found, the magnetic corrections to the polarizability tensor αij of the RE-ion should be determined. Therefore, it is necessary to calculate the permeability tensor components δεij and define the values of refraction indexes n// and n⊥ from the Fresnel equations (see, for example [8]). It is well known that within the visible and infrared spectral ranges the magnetic-optical properties of RE-compounds are mainly due to the allowed (in spin and parity) electric dipolar (ED) transitions in RE-ions. It is important to note that in the first approximation one can neglected by the splitting of the “mixed” 4f (n − 1) 5d configuration levels (the so-called Judd-Ofelt approximation [14,15]) on which (or from which) the allowed optical 4f (n) → 4f (n − 1) 5d transitions (sometimes named by the f→ d transitions) are occuring. In this case, the calculation of magnetic corrections to the polarizability tensor of the RE-ion with L ≠ 0 leads to the following expression for the quadratic magneto-optical contribution to the dielectric permeability of the crystal [6]: i δεαβ = ab ∑ 〈Qαβ (Lˆ ) 〉
(3)
i i Qαβ (Lˆ )
= (1/2)(Lˆ α Lˆ β + Lˆ β Lˆ α ) − (1/3) L (L + 1) is the quadruple where moment operator component of the RE-ion 4f-shell, Lˆ k is the operator of the angular momentum of the RE-ion, angle brackets mean the averaging of the quadruple moment operator of the RE-ion on the wave2 2 functions of the ground state, b = (2/3π ) N [(n02 + 2) /3] (n 0 is the refractive index, N is the number of RE-ions per unit volume), and the summation is carried out for all six non-equivalent positions (i) of the RE-ion in the garnet structure. From expression (3), its detailed consideration is given in Appendix, one can find the following expression for the phase shift caused by the MLB in TbGG and represented by the approximation of the “free REion" (see also formula (A7) in Appendix): β0 (Tb3 +) = 15.146
μ H 2 N (n¯ 2 + 2)2 ωω (erfd )2 ⎛ B ⎞ ⎡ 2 0 2 ⎤ ⎥ 9nπc ¯ ℏ ⎝ kB T ⎠ ⎢ ⎣ (ω0 − ω ) ⎦ ⎜
⎟
22
(4)
3+
ions per unit of where N = 1.26 × 10 is the number of the Tb volume (cm−3) in TbGG, n¯ = 1.7 is the refractive index of garnet, μ B is the Bohr magneton. For the comparison with experimental data it is convenient to rewrite eq. (4) in the following simple form:
ωω β0 (rad/ cm) = K ⎡ 2 0 2 ⎤ ⎢ (ω0 − ω ) ⎥ ⎦ ⎣
(5)
where the coefficient K is equal to K = 0.1256 at T = 90 K and H = 6.5 kOe.
→ → → → → Δn ⎛⎜H //[110], k //[001] ⎟⎞ = Δn ⎜⎛H //[111], k ⊥ H ⎞⎟ ⎝ ⎠ ⎝ ⎠
4. Experimental results and discussion
→ → → → 1 ⎧ ⎛→ Δn ⎛⎜H //[110], k //[11¯0] ⎟⎞ = Δn ⎜H //[001], k ⊥ H ⎟⎞ ⎨ 2 ⎠ ⎝ ⎠ ⎩ ⎝
It is well-known that the trivalent terbium ions occupy the sites of D2 symmetry (so called c-places of orthorhombic symmetry [9]) in the garnet structure. As the Tb3+ ion has an even number of equivalent 4felectrons, the crystal field (CF) of D2 forms the (2J + 1) non-degenerate energy (Stark) levels for each isolated multiplet of Tb3+ [9,10]. However, especially within the multiplets with large J, some of these Stark levels are closely-spaced states. In the presence of an external magnetic field H, the wavefunctions of two such closely-spaced Stark levels are “mixed” and these levels can form what is called a “quasi-doublet” state [9,10]. According to the modern theoretical representations [8,10], a main source of the magnetooptical effects appeared in the garnet
→ → ⎫ → + Δn ⎛⎜H //[111], k ⊥ H ⎟⎞ ⎝ ⎠⎬ ⎭
(6)
→ → → → Δn ⎛⎜ k ⊥ (110]), H //[001] ⎞⎟ = Δn ⎛⎜ k ⊥ (100), H //[001] ⎞⎟ ⎝ ⎠ ⎝ ⎠ However, according to the authors [6], this approach is not always applicable and is valid only for so-called S-ions (with L = 0) [9] at sufficiently high temperatures and not very strong magnetic fields. 105
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Fig. 2. The unit cell of terbium-gallium garnet. Three cationic positions in the different environment of oxygen ions are shown.
Fig. 3. The temperature dependence of MLB in TbGG at λ = 575 nm in the field H = 6.5 kOe. Inset: The functional dependence of MLB on the square of the inverse temperature demonstrates the linearity according to expression (4) (or (A7)).
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(1/ β0) × (1/ λ ) on square of the inverse wavelength 1/ λ2 (see the inset to Fig. 4). The coefficient K and frequency ω0 found in a similar way (from the experimental data given in Fig. 4) are equal to K = 46.7 (min) and ω0 = 5.71 × 1015s−1 (λ 0 = 330 nm), respectively, for the specified thickness of TbGG sample under study. On the other hand, according to formula (5), the coefficient K is equal to K = 59.46 (min) for the same thickness of the sample of the P3-garnet, which is in good agreement with the experimental value of an order of magnitude. It is important to note that the determined “effective” frequency ω0 for the f→ d transition in RE-ions of Tb3+ in TbGG differs from the similar value defined by direct measurements of optical absorption observed at ω0 = 7.16 × 1015s−1 (λ 0 = 263 nm) in thin films of gallium garnet terbium-yttrium within the ultraviolet spectral range [24]. In addition, the significant difference in the “effective” frequencies was found also by comparing their values obtained from our work and measurements of the Faraday effect in TbGG within the visible spectral range ω0 = 7.59 × 1015s−1 (λ 0 = 248 nm [25]). In our opinion, the nature of this difference can be explained as follows. It is known that there is a spin-forbidden electric dipolar transition of the type 4f (8) (7 F6 ) → 4f (7) 5d (9D) within the spectral range of 320–330 nm (5.9 × 1015 - 5.7 × 1015 s−1). It is interesting to note that this transition was observed earlier both in the absorption spectra of terbium-yttrium gallium garnets and in the luminescence excitation spectra of other crystals [26,27]. Apparently, considerable distortions of the crystal lattice caused by the large magnetostriction of the terbium garnet at low temperatures (as compared with other RE-garnets [6]) lead to “mixing” of the states of the 9D term with “optically resolved” states of the 7D term. In turn, the appearance of such a mechanism of states “mixing” leads to a significant contribution of electric dipolar transitions with eigen frequences shifted to the near ultraviolet spectral region to the MLB of terbium-gallium garnet. Thus, the results of the above studies show that in the extreme case of sufficiently high temperatures (≥100 K) the behavior of the spectral, temperature, and field dependences of MLB rare-earth garnet (in particular, TbGG) can be well described by a simple approximation of the “free RE-ion". In other words, in this case the phenomenological approach based on the theory of even magnetic effects of Akulov [22] can be quite realistic. In this approach, the MLB in paramagnets can be represented as [8,23]:
As it follows from the experimental data we obtained for the REgarnet TbGG within the temperature range 90–300 K, MLB changes quadratically both with a magnetic field H and with a decrease in temperature T. More specifically, the MLB value grows inversely proportional to the square of temperature (see also inset in Fig. 3), in accordance with expression (4). On the one hand, the obtained result shows that the ground state (“quasi-doublet”) of the 7F6 multiplet in the Tb3+ ion makes a dominant contribution to the formation mechanism of the temperature dependence of MLD in TbGG within the temperature range 90–300K. Indeed, if a “quasi-doublet” is the ground state of the non-Kramers RE-ion, then the magnetization MR of the two-level REsubsystem is inversely proportional to temperature (i.e. MR ∼ H ) [9]. T On the other hand, the measurement results of the magnetic properties of terbium-gallium garnet for the temperatures 78–300 K [11] showed that within this temperature range the TbGG magnetization within good accuracy can be described by the “free RE ion” approximation, which often uses for the simplest description of temperature dependence for the RE - subsystem magnetization in the garnet structure. Thus, the results of comparing the measurement results of MLD and magnetization in terbium-gallium garnet indicate that it is possible to use the “free RE-ion” approximation for description of the special features of the temperature dependence of MLD in RG garnet. The values of Δn for the three investigated crystallographic directions in terbium gallate TbGG and for the wavelength λ = 575 nm at T = 90 K in the field H = 6.5 K are presented in Table 1. Within the accuracy of experiment (∼3–5%) for them the relations (6) arising from the MLB phenomenological consideration are realized; they connect the values of Δn measured at the orientation of the external field in the plane (110) along the main crystallographic directions of the garnet [6]. It is important to note that the natural crystallographic birefringence caused by the residual growth deformation, mosaicity and other defects of the crystal structure of the sample under consideration was sufficiently small and the value of Δn within the experimental error did not change practically when the sample was rotated by 1800 around the axis perpendicular to its plane. Let us now consider the spectral dependence of MLB in TbGG measured within the wavelength range 480–630 nm at T = 90 K in the external magnetic field H = 6.5 kOe. An external magnetic field H was oriented along the crystallographic direction H//[001]. These measurements were performed using only one non-chromatic Senarmont compensator with the phase shift equal to ≈900 at a wavelength of 580 nm. In this case taking into account the non-achromicity of the phase compensator, we found that the error of MLB measurements did not exceed ∼5% at the boundary wavelengths of the spectral range, i.e. did not exceed the experimental error of our measurements. The dependence of phase shift β0 caused by the MLB in terbium gallate-garnet on frequency factor ωω0 /(ω02 − ω2) is shown in Fig. 4. On the one hand, this spectral dependence has a linear character, which agrees with formula (5). On the other one, it makes possible to compare the experimentally found coefficient K with the theoretically calculated values, which will serve as a criterion for justifying the theoretical consideration of MLB in TbGG under the approximation of the “free REion" performed in the Appendix of the paper. At the same time, the “effective” frequency ω0 of the f→ d electric dipolar transitions in the RE-ions of Tb3+, which is responsible for the occurrence of MLB in terbium gallate-garnet, can be found from the functional dependence of
Δn (H , T ) = A⋅M 2 (H , T )
Δn (H , T ) = A⋅χ 2 (T )⋅H 2
5. Conclusion The results of the present study of the MLB orientational and temperature dependences in TbGG within the visible spectral range complemented by the theoretical consideration allow the following conclusions to be made:
TbGG
1. The results of the above-mentioned magneto-optical experiments in TbGG have confirmed the Akulov's relations (for the quadratic magnetic effects) arising from the phenomenological consideration of MLB for the values of Δn measured at the orientation of the external field in the plane (110) along the main crystallographic directions of the garnet crystal.
Δn × 107 //[001]
H//[111]
H//[11¯0 ]
12.0
4.4
8.1
(8)
Moreover, for cubic crystals the susceptibility χ is isotropic where → the ratios (6) should be fulfilled under the condition k ⊥ (110) .
Table 1 MLB of the investigated garnet in the field H = 6.5 kOe at T = 90 K at a wavelength λ = 575 nm. RGG
(7)
where M is the magnetization; A is the phenomenological coefficient depending on the crystallographic direction and wavelength of the incident light. In the region of weak magnetic fields and not very low temperatures where M = χH ( χ is the magnetic susceptibility) expression (7) is transformed into the form:
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Fig. 4. Linear dependence of the phase shift β0 on frequency factor ωω0 /(ω02 − ω2) in TbGG at T = 90 K in the external field H = 6.5 kOe. Inset: the functional dependence (1/ β0 ) × (1/ λ ) on 1/ λ2 .
Declaration of interests
2. The comparison of the results of magnetic and magneto-optical studies show that in the temperature range 90–300 K the behavior of the spectral, temperature, and field dependences of MLB for TbGG can be described within good accuracy by the approximation of the "free RE-ion". 3. Found from comparison of the absorption spectra of terbium-yttrium garnets with the results of the MLB spectral dependence analysis, the difference between “effective” frequencies of the allowed f→ d electric dipolar transitions in RE-ions of Tb3+ ions can be explained by the considerable distortions of the crystal lattice of TbGG. The similar crystal lattice distortions caused by the significant magnetostriction of TbGG at low temperatures lead to the "mixing" of the states of the 9D term with the "optically resolved" states of the 7D term. As a result, such a mechanism of states "mixing" will leads to a significant contribution of the f→ d transitions with eigen frequences shifted to the near ultraviolet spectral region to the MLB of terbium-gallium garnet.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests.
Acknowledgements This work was supported by the International Cooperation Program of the Ministry of Science and Technology of China (Grant No. 2015DFR00720), the Wuhan Municipal Science and Technology Bureau grant No. 2017030209020250, and by the State Scientific Project of the Republic of Uzbekistan (Grant No. OT-F2-09).
Appendix In the expression (3) the coefficient is proportional to the oscillator force of the allowed f→ d optical transition and equal to [7]: 2
a=
1 ⎡ 2ω0 ⎤ (erfd )2 ∑L′ (2L + 1) ℏ(ω02 − ω2)
⎣
⎦
(L// n1// L′⎞⎟ F (L′, L) ⎠
(A1) 2
(L// d (1)// L′)2 =
L1
F (L′, L) = −
3 L L1⎫ 2 1⎬ ⎭ ⎩
∑ 3n (G SSL′L′)2 (2L + 1)(2L′ + 1) ⎧⎨ L′
(A2)
δ L′, L − 1 δ L′, L δ L′, L + 1 + − L (2L − 1) L (L + 1) (L + 1)(2L + 3)
where ω0 is the “effective” frequency of the f→ d transitions; ω is the light frequency; rfd = (4f / r /5d ) is the radial integral (for the ion Tb3+ rfd = 3.8 × 10−9 cm [15]); L′ is the orbital quantum number for the “optically resolved” terms of the 4f (n − 1) 5d configuration of RE-ion; GSSL = 1/ 7 is 1 L1 the coefficient of fractional parentage [16,17]; L1, S1 are the quantum numbers of the ground term of the “parent ion” [16,17]; (L// n1// L′) is the 108
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j j j matrix element of the unit vector [16]; ⎧ 1 2 3 ⎫ is the 6j-symbol [16,18]; δL, L ± 1,0 is the Kronecker's symbol. ⎨ l ⎭ ⎩ 1 l2 l3 ⎬ It should be noted that for the allowed 4f (8) → 4f (7) 5d transition in the Tb3+ ion the 8S term is the lowest one within the 4f (7) configuration 3+ associated with so-called «parent» ion Gd [16]. Then, adding its orbital L1 = 0 and spin S1 = 7/2 momentum with analogous momentum of “valence” d - electron by common summation rules of momentum [16–18], it is easy to show that the “optically resolved” term of the mixed excited 4f (7) 5d configuration is the term of 7D (L′ = 2, S′ = 3) [10,19]. Thus the subsequent consideration will be based on a dominating contribution of allowed electro dipolar 4f (8) (7 F6 ) → 4f (7) 5d (7D) transition to the MLB in TbGG. As shown in Ref. [8], for the incident light frequency within the optical range a gyrotropic component of the εαβ tensor can be neglected. In this → case, for the coordinate system with the axis z // H , the magnitude of the double light refraction Δn is connected with the components of the dielectric penetrability tensor δεij by a relatively simple expression Δn = (3/4) n0−1 δεzz . Then the expression (1) for the phase shift β0 in the “free RE-ion” approximation becomes as follows [7]: β0 = 9(n 0 λ )−1⋅ab⋅〈Qzz 〉 = C′⋅〈Qzz 〉
(A3)
3 (n 0 λ )−1⋅ab . 2
where С′= 6 × Averaging the operator Qzz by the ground state/ LSJm> for the Tb3+ RE-ion, one can obtain [7]:
β0 = C′⋅(L // n1// L)2 ∑ β′m (J )
(A4)
m
In this approximation A4 can be rewritten for β′m as follows (see equation (9) in Ref. [7]):
β′m (J ) =
ρm =
L S J⎫ (2J + 1) ⋅(−1) L + S + J ⎧ 2 J L⎬ ⎨ ⎭ ⎩
∑ m
[3m2 − J (J + 1)] (2J − 1) J (J + 1)(2J + 3)
ρm
(A5)
exp(−mgμB H / kB T ) ∑m exp(−mgμB H / kB T )
where g = 1.5 is the Lande factor of the ground 7F6 multiplet of the Tb3+ ion. The value of β0 is a phase shift caused by MLB in the absence of Van Vleck's “mixing” of the neighboring multiplets wavefunctions by the external magnetic field H [7]. Note that in the above-mentioned work [7] the influence of the magnetic “mixing” of the RE-ion multiplets (particularly, for Sm3+ ion) on the formation mechanisms of quadratic magneto-optical effects in RE-garnets is considered in detail. Since the parameter x = mgμB H / kB T at high temperatures (and small magnetic fields) satisfies the condition x < < 1, the expression for the Boltzmann population can be decomposed into a Taylor series up to members of the second order of smallness. After the summation in expression (A5) by index m is made, it is easy to find that:
∑m β′m (J ) =
= 6.493 × 10−3 ⎡ ⎣ Since terms)
1 2
m =+J
∑ m =−J
[3m2 − J (J + 1)]
L S J ⎫ m =+J (2J + 1) ⋅⎧ ∑ 2 J L ⎬ m =−J ⎨ ⎭ ⎩
m =+J ∑m =−J
J (J + 1) gμB H 2 ⎤ kB T ⎦
[3m2
(2J − 1) J (J + 1)(2J + 3)
⎡1 − ⎣
mgμB H kB T
+
1 2
(
mgμB H 2 kB T
)
+ ...⎤= ⎦
μ H 2
= 25.77 ⎡ kB T ⎤ ⎣ B ⎦
(A6)
− J (J + 1)] = J (J + 1)(2J + 1) − J (J + 1)(2J + 1) =
m =+J 0∑m =−J
[3m2
− J (J +
mgμ H 1)] k BT B
= 0 (because of odd character of the sum
mgμB H ⎞2 J (J + 1) gμB H ⎤2 [3m2 − J (J + 1)] ⎛ = 1.755 ⎡ ⎢ ⎥ kB T ⎝ kB T ⎠ ⎣ ⎦ ⎜
⎟
Using expression (A4) and substituting formulas (A1) and (A5), we can calculate the expression for the phase shift caused by the MLB in TbGG and write it in the approximation of the “free RE-ion” as follows:
β0 (Tb3 +) = 15.146
μ H 2 N (n¯ 2 + 2)2 ωω (erfd )2 ⎛ B ⎞ ⎡ 2 0 2 ⎤ ⎥ 9nπc ¯ ℏ ⎝ kB T ⎠ ⎢ ⎣ (ω0 − ω ) ⎦ ⎜
3+
where N is the number of the Tb
⎟
(A7)
ions per unit of volume, n¯ is the refractive index of garnet, μ B is the Bohr magneton.
[11] U.V. Valiev, G.S. Krinchik, S.B. Kruglyashov, R.Z. Levitin, K.M. Mukimov, V.N. Orlov, B. Yu Sokolov, Solid State Phys. 24 (1982) 2818. [12] J. Badoz, M. Billardon, J.C. Canit, M.F. Russel, J. Optic. 8 (1977) 373. [13] S.N. Jasperson, S.E. Schnatterly, Rev. Sci. Instrum. 40 (1969) 761. [14] B.R. Judd, Phys. Rev. 127 (1962) 750. [15] G. Ofelt, J. Chem, Physics 37 (1962) 511. [16] I.I. Sobelman, Introduction to the Theory of Atomic Spectra, Pergamon, New York, 1972. [17] B.G. Wybourne, Spectroscopic Properties of Rare-earth, (1965) New York. [18] D. Varshalovich, A. Moskalev, V. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1989. [19] U.V. Valiev, A.I. Popov, B. Yu Sokolov, Optic Spectrosc. 61 (1986) 714. [20] N.P. Kolmakova, S.V. Koptsik, G.S. Krinchik, V.N. Orlov, A. Ya, Sarantsev. Sov. Phys. Sol. St. 32 (1990) 821. [21] John B. Gruber, Dhiraj K. Sardar, Raylon M. Yow, U.V. Valiev, A.K. Mukhammadiev, V. Yu Sokolov, K. Lengyel, I.S. Kachur, Valeriya G. Piryatinskaya, Zandi Bahram, J. Appl. Phys. 101 (2007) 023108. [22] N.S. Akulov, Ferromagnetism, Gostekhizdat, Moscow-Leningrad, 1939 [in Russian].
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J.F. Dillon Jr., J. Appl. Phys. 29 (1958) 1286. R.V. Pisarev, I.G. Sinii, G.A. Smolenskii, JETP Lett 9 (64) (1969) 172. R.V. Pisarev, I.G. Sinii, G.A. Smolenskii, Sov. Phys. JETP 30 (3) (1970) 404. J. Ferre, G.A. Gehring, Rep. Prog. Phys. 47 (1984) 513. J.C. Suits, B.E. Argyle, M.F. Freiser, J. Appl. Phys. 37 (1966) 1391. N.F. Vedernikov, A.K. Zvezdin, R.Z. Levitin, A.I. Popov, Sov. Phys. JETP 66 (6) (1987) 1233. O.A. Dorofeev, A.I. Popov, Phys. Solid State (St.-Peter.) 35 (6) (1993) 740. A.K. Zvezdin, A.V. Kotov, Modern Magnetooptics and Magnetooptical Materials, IOP Publishing, Bristol and Philadelphia, 1997, p. 386. A.K. Zvezdin, V.M. Matveev, A.A. Mukhin, A.I. Popov, Rare earth ions in magnetically ordered crystals, [in Russian] Nauka, Izdatel, 1985 Moscow. U.V. Valiev, J.B. Gruber, G.W. Burdick, Magnetooptical Spectroscopy of the RareEarth Compounds: Development and Application, Scientific Research Publishing, USA, 2012, p. 139.
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[23] Zh Sh Siranov, B. Yu Sokolov, Optic Spectrosc. 81 (6) (1996) 926. [24] Uygun V. Valiev, John B. Gruber, Gary W. Burdick, Anvar K. Mukhammadiev, Dejun Fu, O. Vasiliy, Pelenovich. Opt. Mater. 36 (2014) 1101. [25] U.V. Valiev, T.V. Virovets, R.Z. Levitin, K.M. Mukimov, B. Yu Sokolov,
M.M. Turganov, Optic Spectrosc. 57 (4) (1984) 757. [26] U.V. Valiev, A.A. Klochkov, V. Nekvasil, Optic Spectrosc. 75 (1) (1993) 33. [27] T. Hoshina, S. Kuboniwa, J. Phys. Soc. Jap. 32 (1972) 771.
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Optical Materials journal homepage: www.elsevier.com/locate/optmat
Magnetic linear birefringence of light in Tb3Ga5O12 a
b
c
T b
Rakhim Yu Rakhimov , Uygun V. Valiev , Denis N. Karimov , Ramil R. Vildanov , Dejun Fu
a,∗
a Key Laboratory of Artificial Micro- and Nano-Structures of the Ministry of Education of China and Hubei Key Laboratory of Nuclear Solid Physics at the School of Physics and Technology, Wuhan University, Wuhan, 430072, China b Faculty of Physics, National University of Uzbekistan, Tashkent, 100174, Uzbekistan c Federal Scientific Research Centre «Crystallography and Photonics», Russian Academy of Sciences, Moscow, 119333, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Magnetic linear birefringence Terbium-gallium garnet Akulov's relation
The spectra of the magnetic linear birefringence (MLB) in the terbium-gallium garnet TbGG (Tb3Ga5O12) have been studied within the visible spectral range for temperature T = 90 K. Analysis of the temperature and spectral dependences of MLB has made it possible to determine the “effective” frequencies of the allowed f→ d electric dipolar transitions in rare-earth (RE) ions of Tb3+ ions responsible for the formation of quadratic magnetooptical effect in this garnet. The results of the performed magneto-optical experiments in TbGG have confirmed the Akulov's relations (for the quadratic magnetic effects) arising from the phenomenological consideration of MLB for the values of Δn measured at the orientation of the external field in the plane (110) along the main crystallographic directions of the garnet crystal. From analysis of the experimental data on MLB was obtained that for the TbGG MLB changes quadratically with the field within the temperature range 90–300 K. At the same time with a decrease in temperature, its value grows inversely proportional to the square of temperature. A certain difference was also found in the “effective" frequencies of the allowed transitions in Tb3+ ions of the garnet, which is determined from the optical absorption spectra of TbGG on the one hand, and from the analysis of the spectral dependence of MLD on the other one. It has been assumed that this difference can be due to significant distortions of the TbGG crystal lattice caused by this significant magnetostriction at low temperatures, leading to the "mixing" of the 9D term states with the "optically resolved" states of the 7D term.
1. Introduction The magnetic linear birefringence (MLB) observed in the rare-earth ferrite-garnets (REFG) [1–3] and in a number of other crystals with different magnetic structures [4,5] is quite large, Δn ∼10−5–10−3. An investigation of this phenomenon in magnetically ordered crystals is of great interest since it gives a unique possibility of associating the optical characteristics of crystals with magnetic ones. Establishment of such a relation is important for studying the spin dependence of the crystal polarizability. Such a relation can be used intensively for the study, by optical methods, of the magnetic structure of crystals, magnetization of sublattices, magnetic phase transitions and other phenomena. At the same time, in the late 20th century, significant interest has been attracted to the MLB research in the paramagnetic rare-earth (RE) garnets [6,7]. By the numerous experimental studies carried out with the REFG [1–4], it was found that MLD in them is largely determined by a contribution of the subsystem of RE-ions and varies greatly in the transition from one ion to another. In particularly, in the REFG series, MLB can vary with temperature
∗
by various ways (up to the change of a sign of the effect). The temperature dependences of the MLB for different crystallographic directions, and even in the same garnet, can differ significantly from each other (see, for example [3,4]). The phenomenological theory operating with the quadratic dependence of MLB on magnetization experiences significant difficulties in the interpretation of the temperature, field and orientation dependencies of MLB in REFG [2–4]. Indeed, for their detailed analysis within the microscopic approach, it is necessary to know the wavefunctions and energy spectra of the rare-earth ion in garnet [6–8]: those parameters are now insufficiently studied. A complex magnetic structure of REFG is formed by a superposition of several magnetic sublattices each of which contributes to the MLD. In turn, the temperature-dependent contribution of a single sublattice is determined by its orientation relative to the crystal axes and direction of the external magnetic field, which greatly complicates the consideration of its contribution to MLD. Therefore, the reason for the difference in the temperature dependences of MLB in the REFG series is still not sufficiently clear. At the same time, in the systems simpler than the REFG, i.e. the rare-earth paramagnetic garnets (gallates and aluminates), a
Corresponding author. E-mail addresses: [email protected], [email protected] (D. Fu).
https://doi.org/10.1016/j.optmat.2018.11.022 Received 22 July 2018; Received in revised form 8 November 2018; Accepted 15 November 2018 0925-3467/ © 2018 Published by Elsevier B.V.
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Passing through the Faraday cell (FC) and analyzer (A), the light was focused on the input slit of a high resolution double diffraction monochromator (MDR model 23, LOMO, Russia). After that the passing light is incident on a cathode of the photomultiplier (PMT). The monochromator (M) have a spectral resolution generally better than 0.05 nm over the wavelength range under study. The state of light polarization was modulated with a frequency of Ω = 80 Hz by the Faraday cell [4,6] made from a terbium glass bar (10 cm in length). The signal was registered by a narrow-band (selective) amplifier (SA) and synchronous detector (SD). Magnetic birefringence occurs in a crystal as a result of the fact that → → → → → two light waves polarized with E // H and E ⊥ H , where E is the electric → vector of the light wave and H is the external magnetic field, propagating in the crystal perpendicular to the magnetization have different phase velocities. Plane-polarized light passed through the crystal has the electric vector oriented at an angle of 45° to the direction of the external magnetic field during the magnetic birefringence measure→ ments. The E components parallel and perpendicular to the external magnetic field are then equal in amplitude and the birefringence is only associated with the difference in the phase velocities of these components. The relationship between the phase difference β and the difference between the refractive indices Δn = n// − n⊥ is given by formula [4]:
similar problem has been resolved in the framework of microscopic representations developed in Refs. [6–8]. Indeed, according to [6–8], in the case of rare-earth magnetics, the MLB is determined by a RE-ion quadrupole momentum Q induced by an external magnetic (or effective exchange) field. In this case, the features of the MLB behavior are determined both by the electronic structure of the magnetoactive ions formed in the low-symmetry (symmetry D2) crystal environment of the RE-ion and by the changes in the quadrupole moment of the RE-ion. These changes in the Q arise from the Van Vleck's “mixing” [9] of the wavefunctions of the sublevels of the ground multiplet by the external magnetic field H. In this case, the MLB behavior also depends strongly on a character of the crystal field (CF) splitting of the Stark structure of the ground multiplet of the RE-ion in the garnet crystal. As shown in Ref. [6], the splitting of RE-ion multiplets in the crystal field of garnet-gallate (or aluminate) leads to the appearance of doublet levels for the so-called Kramers ions (Dy3+, Er3+, Yb3+) or singlet levels for non-Kramers ions (Eu3+, Tb3+, Tm3+, Ho3+), according to Kramers theorem [9,10]. This important circumstance is associated with the emergence of field and the temperature dependences of MLD are significantly different from similar dependences predicted in the framework of the traditional theory of quadratic magneto-optical phenomena [2,4,5]. In addition, an investigation of the MLB spectral dependence for REgarnets can also serve as an additional criterion for the adequacy of the theoretical constructions developed in Refs. [6,7]. The latter circumstance is very relevant for the terbium-gallium garnet Tb3Ga5O12 (TbGG) since the theoretical analysis of the field, spectral and temperature dependences of MLB in TbGG, measured in present work, can be substantially simplified in the limiting case of sufficiently high temperatures (≥100 K). The reason for this simplification is that, according to magnetic measurements [11], the magnetization of TbGG RE-garnet can be very well described by the approximation of a “free RE-ion” within the temperature range of 78–300 K. Therefore, a special character of the behavior of the TbGG magnetic properties [11] allowed us to interpret the data of MLB measurements in TbGG within the temperature range 90–300 K by the so-called approximation of a “free Tb3+-ion".
β=
2π Δn⋅l λ
(1)
here λ is the light wavelength in vacuum, l is the crystal length. If one of the axes of the Senarmont compensator is parallel to the transmission plane of the polarizer (P) then the angle θ between the transmission planes of the polarizer (P) and the analyzer (A) is close to 90 deg, i.e. θ = 900 ± Δ. Then the main component I (Ω) of the intensity of the light I incident on the photomultiplier photocathode will be as follows:
I (Ω) =
I0 [2J1 (2α 0 )⋅sin(2θ + β )⋅sin Ωt ] 2
(2)
where I0 is the light intensity at the analyzer; J1 (x ) is the first-order Bessel function [12,13]; α 0 = 1 − .1. 50 is the amplitude of rotation angle provided by the Faraday cell; Ω is the modulation frequency. It should be noted that in this method of measurement so-called “zeromethod” of MLB measurement can be realized. In this case, the signal I (Ω) induced by a magnetized sample can be compensated by the rotation of an output analyzer placed after the Senarmont compensator at an angle equal to Δ. As a result, the rotation angle of the analyzer is equal to half of the phase difference β caused by MLB, according to the β relation Δ = ± 2 . Single crystal of terbium-gallium garnet TbGG was grown by a method of spontaneous crystallization from a solution-melt (Czochralski method) and provided by Prof. B.V. Mill from Moscow State University (Russia). A sample was oriented by the XRD method and cut in the
2. Measurement technique and samples An experimental setup for the measuring of the magnetic linear birefringence spectra within the visible region was used. A 100W halogen lamp was utilized as a light source (see Fig. 1). The light from the halogen lamp (S) is collected by a quartz condenser (L) and goes through a polarizer (P) to a crystal (TbGG) mounted in a cryostat with controllable temperature. The cryostat is placed in a gap of the electromagnet (EM) with the magnetic field (H = 6.5 kOe) perpendicular to the direction of light propagation. The phase difference that the light acquired in the crystal was measured with an optical Senarmont compensator (λ/4) placed between a crossed polarizer and analyzer.
Fig. 1. S is the light source, L is the quartz lens, P is the polarizer, EM is the electromagnet, PS is the power supply for EM, TbGG is the garnet sample, λ/4 is the Senarmont compensator, FC is the Faraday cell, PA is the power amplifier for FC, G is the generator, A is the analyzer, PMT is the photomultiplier, A is the preliminary amplifier (with high resistance), SA is the selective amplifier, SD is the synchronous detector.
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crystals doped with non-Kramers ions is “mixing” of the wavefunctions of the closely located non-degenerate Stark singlets in the “quasidoublets” by an external field H. Indeed, from an analysis of the transverse and longitudinal Zeeman effect observed in absorption of the 5 D4 multiplet the authors of [20,21] identified the ground state of the Tb3+ ion in TbGG as a “quasidoublet” consisting of two lowest nondegenerate Stark levels. According to the CF calculations [21], the wavefunctions of this “quasi-doublet” can be described by “pure”/ J,m > states characterized by the maximum values of the angular momentum projections m = ± 6 and making the largest contribution to the average magnetic moment of the PE subsystem. Therefore, it is natural to expect that in the formation mechanism of the temperature dependence of both magnetization and MLD in terbium-gallium garnet, this quantum state will plays a main role. Since the presented-here studies consider not only the temperature and spectral dependences of MLB in TbGG, but also orientation ones, it seems appropriate to make a few detailed comments concerning the features of the crystallographic structure of RE-garnets. Terbium-gallium garnet is crystallographically cubic and belongs to the space group Ia3d. There are eight formula units per cell. The unit cell of terbiumgallium garnet crystal, containing 160 atoms, is shown in Fig. 2. The cations (Tb3+ and Ga3+) are surrounded by oxygen atoms in the various symmetry in the garnet unit cell and crystal structure of this compound can be represented in the coordination polyhedron form of the interconnected dodecahedrons, octahedrons and tetrahedrons with common oxygen atoms. The Tb3+ ions occupy (24c) positions and each Tb3+ ion is coordinated with eight oxygen ions in the so-called dodecahedral site (distorted cube form). The Ga3+ ions occupy the (16a) positions in octahedrons and tetrahedral (24d) positions in the crystal lattice. The oxygen ions O2− occupy the (96h) positions and each oxygen ion is a part of two dodecahedrons, one octahedron and one tetrahedron. The (110) crystal plane and positions of the specific crystallographic [001], [110] and [111] directions are schematically shown in Fig. 2. These specified directions are of interest for orientation-dependent magneto-optical investigation in the terbium-gallium garnet. From Fig. 2 it is also clearly seen that the crystallographic axes of the type [001] and [110], as well as the axes [110] and [111], are mutually perpendicular to each other, respectively. But then it can be expected that in the MLB the sum of changes in the refractive indices for the axes [001], [111] and [110] of the garnet cubic crystal will be equal to zero [6]. This statement can follows from the law of Akulov for the even (i.e. quadratic) magnetic effects as this law states that “the summation of the results of measurements of an even magnetic effects for the three arbitrary mutually perpendicular directions is equal to zero [22]". Indeed, for a number of the cubic crystals of rare-earth garnet, i.e. gallates RGG (where R = Eu, Tb, Er, Tm) and aluminates RAG (R = Dy, Ho, Er, Yb), the following relations for MLB at the different → → orientations of vectors H and k were found in Ref. [6] and experimentally confirmed in Ref. [23]:
crystallographic plane [110], its surface was polished by diamond pastes using slowly diminishing grain size (down to ∼ 1 mkm). The thickness of the TbGG sample used in the magneto-optical measurements was ∼1.5 mm. 3. Some features of the theoretical description of MLB in TbGG As shown earlier in Refs. [6–8], for the phase shift β arising from the MLB to be found, the magnetic corrections to the polarizability tensor αij of the RE-ion should be determined. Therefore, it is necessary to calculate the permeability tensor components δεij and define the values of refraction indexes n// and n⊥ from the Fresnel equations (see, for example [8]). It is well known that within the visible and infrared spectral ranges the magnetic-optical properties of RE-compounds are mainly due to the allowed (in spin and parity) electric dipolar (ED) transitions in RE-ions. It is important to note that in the first approximation one can neglected by the splitting of the “mixed” 4f (n − 1) 5d configuration levels (the so-called Judd-Ofelt approximation [14,15]) on which (or from which) the allowed optical 4f (n) → 4f (n − 1) 5d transitions (sometimes named by the f→ d transitions) are occuring. In this case, the calculation of magnetic corrections to the polarizability tensor of the RE-ion with L ≠ 0 leads to the following expression for the quadratic magneto-optical contribution to the dielectric permeability of the crystal [6]: i δεαβ = ab ∑ 〈Qαβ (Lˆ ) 〉
(3)
i i Qαβ (Lˆ )
= (1/2)(Lˆ α Lˆ β + Lˆ β Lˆ α ) − (1/3) L (L + 1) is the quadruple where moment operator component of the RE-ion 4f-shell, Lˆ k is the operator of the angular momentum of the RE-ion, angle brackets mean the averaging of the quadruple moment operator of the RE-ion on the wave2 2 functions of the ground state, b = (2/3π ) N [(n02 + 2) /3] (n 0 is the refractive index, N is the number of RE-ions per unit volume), and the summation is carried out for all six non-equivalent positions (i) of the RE-ion in the garnet structure. From expression (3), its detailed consideration is given in Appendix, one can find the following expression for the phase shift caused by the MLB in TbGG and represented by the approximation of the “free REion" (see also formula (A7) in Appendix): β0 (Tb3 +) = 15.146
μ H 2 N (n¯ 2 + 2)2 ωω (erfd )2 ⎛ B ⎞ ⎡ 2 0 2 ⎤ ⎥ 9nπc ¯ ℏ ⎝ kB T ⎠ ⎢ ⎣ (ω0 − ω ) ⎦ ⎜
⎟
22
(4)
3+
ions per unit of where N = 1.26 × 10 is the number of the Tb volume (cm−3) in TbGG, n¯ = 1.7 is the refractive index of garnet, μ B is the Bohr magneton. For the comparison with experimental data it is convenient to rewrite eq. (4) in the following simple form:
ωω β0 (rad/ cm) = K ⎡ 2 0 2 ⎤ ⎢ (ω0 − ω ) ⎥ ⎦ ⎣
(5)
where the coefficient K is equal to K = 0.1256 at T = 90 K and H = 6.5 kOe.
→ → → → → Δn ⎛⎜H //[110], k //[001] ⎟⎞ = Δn ⎜⎛H //[111], k ⊥ H ⎞⎟ ⎝ ⎠ ⎝ ⎠
4. Experimental results and discussion
→ → → → 1 ⎧ ⎛→ Δn ⎛⎜H //[110], k //[11¯0] ⎟⎞ = Δn ⎜H //[001], k ⊥ H ⎟⎞ ⎨ 2 ⎠ ⎝ ⎠ ⎩ ⎝
It is well-known that the trivalent terbium ions occupy the sites of D2 symmetry (so called c-places of orthorhombic symmetry [9]) in the garnet structure. As the Tb3+ ion has an even number of equivalent 4felectrons, the crystal field (CF) of D2 forms the (2J + 1) non-degenerate energy (Stark) levels for each isolated multiplet of Tb3+ [9,10]. However, especially within the multiplets with large J, some of these Stark levels are closely-spaced states. In the presence of an external magnetic field H, the wavefunctions of two such closely-spaced Stark levels are “mixed” and these levels can form what is called a “quasi-doublet” state [9,10]. According to the modern theoretical representations [8,10], a main source of the magnetooptical effects appeared in the garnet
→ → ⎫ → + Δn ⎛⎜H //[111], k ⊥ H ⎟⎞ ⎝ ⎠⎬ ⎭
(6)
→ → → → Δn ⎛⎜ k ⊥ (110]), H //[001] ⎞⎟ = Δn ⎛⎜ k ⊥ (100), H //[001] ⎞⎟ ⎝ ⎠ ⎝ ⎠ However, according to the authors [6], this approach is not always applicable and is valid only for so-called S-ions (with L = 0) [9] at sufficiently high temperatures and not very strong magnetic fields. 105
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Fig. 2. The unit cell of terbium-gallium garnet. Three cationic positions in the different environment of oxygen ions are shown.
Fig. 3. The temperature dependence of MLB in TbGG at λ = 575 nm in the field H = 6.5 kOe. Inset: The functional dependence of MLB on the square of the inverse temperature demonstrates the linearity according to expression (4) (or (A7)).
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(1/ β0) × (1/ λ ) on square of the inverse wavelength 1/ λ2 (see the inset to Fig. 4). The coefficient K and frequency ω0 found in a similar way (from the experimental data given in Fig. 4) are equal to K = 46.7 (min) and ω0 = 5.71 × 1015s−1 (λ 0 = 330 nm), respectively, for the specified thickness of TbGG sample under study. On the other hand, according to formula (5), the coefficient K is equal to K = 59.46 (min) for the same thickness of the sample of the P3-garnet, which is in good agreement with the experimental value of an order of magnitude. It is important to note that the determined “effective” frequency ω0 for the f→ d transition in RE-ions of Tb3+ in TbGG differs from the similar value defined by direct measurements of optical absorption observed at ω0 = 7.16 × 1015s−1 (λ 0 = 263 nm) in thin films of gallium garnet terbium-yttrium within the ultraviolet spectral range [24]. In addition, the significant difference in the “effective” frequencies was found also by comparing their values obtained from our work and measurements of the Faraday effect in TbGG within the visible spectral range ω0 = 7.59 × 1015s−1 (λ 0 = 248 nm [25]). In our opinion, the nature of this difference can be explained as follows. It is known that there is a spin-forbidden electric dipolar transition of the type 4f (8) (7 F6 ) → 4f (7) 5d (9D) within the spectral range of 320–330 nm (5.9 × 1015 - 5.7 × 1015 s−1). It is interesting to note that this transition was observed earlier both in the absorption spectra of terbium-yttrium gallium garnets and in the luminescence excitation spectra of other crystals [26,27]. Apparently, considerable distortions of the crystal lattice caused by the large magnetostriction of the terbium garnet at low temperatures (as compared with other RE-garnets [6]) lead to “mixing” of the states of the 9D term with “optically resolved” states of the 7D term. In turn, the appearance of such a mechanism of states “mixing” leads to a significant contribution of electric dipolar transitions with eigen frequences shifted to the near ultraviolet spectral region to the MLB of terbium-gallium garnet. Thus, the results of the above studies show that in the extreme case of sufficiently high temperatures (≥100 K) the behavior of the spectral, temperature, and field dependences of MLB rare-earth garnet (in particular, TbGG) can be well described by a simple approximation of the “free RE-ion". In other words, in this case the phenomenological approach based on the theory of even magnetic effects of Akulov [22] can be quite realistic. In this approach, the MLB in paramagnets can be represented as [8,23]:
As it follows from the experimental data we obtained for the REgarnet TbGG within the temperature range 90–300 K, MLB changes quadratically both with a magnetic field H and with a decrease in temperature T. More specifically, the MLB value grows inversely proportional to the square of temperature (see also inset in Fig. 3), in accordance with expression (4). On the one hand, the obtained result shows that the ground state (“quasi-doublet”) of the 7F6 multiplet in the Tb3+ ion makes a dominant contribution to the formation mechanism of the temperature dependence of MLD in TbGG within the temperature range 90–300K. Indeed, if a “quasi-doublet” is the ground state of the non-Kramers RE-ion, then the magnetization MR of the two-level REsubsystem is inversely proportional to temperature (i.e. MR ∼ H ) [9]. T On the other hand, the measurement results of the magnetic properties of terbium-gallium garnet for the temperatures 78–300 K [11] showed that within this temperature range the TbGG magnetization within good accuracy can be described by the “free RE ion” approximation, which often uses for the simplest description of temperature dependence for the RE - subsystem magnetization in the garnet structure. Thus, the results of comparing the measurement results of MLD and magnetization in terbium-gallium garnet indicate that it is possible to use the “free RE-ion” approximation for description of the special features of the temperature dependence of MLD in RG garnet. The values of Δn for the three investigated crystallographic directions in terbium gallate TbGG and for the wavelength λ = 575 nm at T = 90 K in the field H = 6.5 K are presented in Table 1. Within the accuracy of experiment (∼3–5%) for them the relations (6) arising from the MLB phenomenological consideration are realized; they connect the values of Δn measured at the orientation of the external field in the plane (110) along the main crystallographic directions of the garnet [6]. It is important to note that the natural crystallographic birefringence caused by the residual growth deformation, mosaicity and other defects of the crystal structure of the sample under consideration was sufficiently small and the value of Δn within the experimental error did not change practically when the sample was rotated by 1800 around the axis perpendicular to its plane. Let us now consider the spectral dependence of MLB in TbGG measured within the wavelength range 480–630 nm at T = 90 K in the external magnetic field H = 6.5 kOe. An external magnetic field H was oriented along the crystallographic direction H//[001]. These measurements were performed using only one non-chromatic Senarmont compensator with the phase shift equal to ≈900 at a wavelength of 580 nm. In this case taking into account the non-achromicity of the phase compensator, we found that the error of MLB measurements did not exceed ∼5% at the boundary wavelengths of the spectral range, i.e. did not exceed the experimental error of our measurements. The dependence of phase shift β0 caused by the MLB in terbium gallate-garnet on frequency factor ωω0 /(ω02 − ω2) is shown in Fig. 4. On the one hand, this spectral dependence has a linear character, which agrees with formula (5). On the other one, it makes possible to compare the experimentally found coefficient K with the theoretically calculated values, which will serve as a criterion for justifying the theoretical consideration of MLB in TbGG under the approximation of the “free REion" performed in the Appendix of the paper. At the same time, the “effective” frequency ω0 of the f→ d electric dipolar transitions in the RE-ions of Tb3+, which is responsible for the occurrence of MLB in terbium gallate-garnet, can be found from the functional dependence of
Δn (H , T ) = A⋅M 2 (H , T )
Δn (H , T ) = A⋅χ 2 (T )⋅H 2
5. Conclusion The results of the present study of the MLB orientational and temperature dependences in TbGG within the visible spectral range complemented by the theoretical consideration allow the following conclusions to be made:
TbGG
1. The results of the above-mentioned magneto-optical experiments in TbGG have confirmed the Akulov's relations (for the quadratic magnetic effects) arising from the phenomenological consideration of MLB for the values of Δn measured at the orientation of the external field in the plane (110) along the main crystallographic directions of the garnet crystal.
Δn × 107 //[001]
H//[111]
H//[11¯0 ]
12.0
4.4
8.1
(8)
Moreover, for cubic crystals the susceptibility χ is isotropic where → the ratios (6) should be fulfilled under the condition k ⊥ (110) .
Table 1 MLB of the investigated garnet in the field H = 6.5 kOe at T = 90 K at a wavelength λ = 575 nm. RGG
(7)
where M is the magnetization; A is the phenomenological coefficient depending on the crystallographic direction and wavelength of the incident light. In the region of weak magnetic fields and not very low temperatures where M = χH ( χ is the magnetic susceptibility) expression (7) is transformed into the form:
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Fig. 4. Linear dependence of the phase shift β0 on frequency factor ωω0 /(ω02 − ω2) in TbGG at T = 90 K in the external field H = 6.5 kOe. Inset: the functional dependence (1/ β0 ) × (1/ λ ) on 1/ λ2 .
Declaration of interests
2. The comparison of the results of magnetic and magneto-optical studies show that in the temperature range 90–300 K the behavior of the spectral, temperature, and field dependences of MLB for TbGG can be described within good accuracy by the approximation of the "free RE-ion". 3. Found from comparison of the absorption spectra of terbium-yttrium garnets with the results of the MLB spectral dependence analysis, the difference between “effective” frequencies of the allowed f→ d electric dipolar transitions in RE-ions of Tb3+ ions can be explained by the considerable distortions of the crystal lattice of TbGG. The similar crystal lattice distortions caused by the significant magnetostriction of TbGG at low temperatures lead to the "mixing" of the states of the 9D term with the "optically resolved" states of the 7D term. As a result, such a mechanism of states "mixing" will leads to a significant contribution of the f→ d transitions with eigen frequences shifted to the near ultraviolet spectral region to the MLB of terbium-gallium garnet.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests.
Acknowledgements This work was supported by the International Cooperation Program of the Ministry of Science and Technology of China (Grant No. 2015DFR00720), the Wuhan Municipal Science and Technology Bureau grant No. 2017030209020250, and by the State Scientific Project of the Republic of Uzbekistan (Grant No. OT-F2-09).
Appendix In the expression (3) the coefficient is proportional to the oscillator force of the allowed f→ d optical transition and equal to [7]: 2
a=
1 ⎡ 2ω0 ⎤ (erfd )2 ∑L′ (2L + 1) ℏ(ω02 − ω2)
⎣
⎦
(L// n1// L′⎞⎟ F (L′, L) ⎠
(A1) 2
(L// d (1)// L′)2 =
L1
F (L′, L) = −
3 L L1⎫ 2 1⎬ ⎭ ⎩
∑ 3n (G SSL′L′)2 (2L + 1)(2L′ + 1) ⎧⎨ L′
(A2)
δ L′, L − 1 δ L′, L δ L′, L + 1 + − L (2L − 1) L (L + 1) (L + 1)(2L + 3)
where ω0 is the “effective” frequency of the f→ d transitions; ω is the light frequency; rfd = (4f / r /5d ) is the radial integral (for the ion Tb3+ rfd = 3.8 × 10−9 cm [15]); L′ is the orbital quantum number for the “optically resolved” terms of the 4f (n − 1) 5d configuration of RE-ion; GSSL = 1/ 7 is 1 L1 the coefficient of fractional parentage [16,17]; L1, S1 are the quantum numbers of the ground term of the “parent ion” [16,17]; (L// n1// L′) is the 108
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j j j matrix element of the unit vector [16]; ⎧ 1 2 3 ⎫ is the 6j-symbol [16,18]; δL, L ± 1,0 is the Kronecker's symbol. ⎨ l ⎭ ⎩ 1 l2 l3 ⎬ It should be noted that for the allowed 4f (8) → 4f (7) 5d transition in the Tb3+ ion the 8S term is the lowest one within the 4f (7) configuration 3+ associated with so-called «parent» ion Gd [16]. Then, adding its orbital L1 = 0 and spin S1 = 7/2 momentum with analogous momentum of “valence” d - electron by common summation rules of momentum [16–18], it is easy to show that the “optically resolved” term of the mixed excited 4f (7) 5d configuration is the term of 7D (L′ = 2, S′ = 3) [10,19]. Thus the subsequent consideration will be based on a dominating contribution of allowed electro dipolar 4f (8) (7 F6 ) → 4f (7) 5d (7D) transition to the MLB in TbGG. As shown in Ref. [8], for the incident light frequency within the optical range a gyrotropic component of the εαβ tensor can be neglected. In this → case, for the coordinate system with the axis z // H , the magnitude of the double light refraction Δn is connected with the components of the dielectric penetrability tensor δεij by a relatively simple expression Δn = (3/4) n0−1 δεzz . Then the expression (1) for the phase shift β0 in the “free RE-ion” approximation becomes as follows [7]: β0 = 9(n 0 λ )−1⋅ab⋅〈Qzz 〉 = C′⋅〈Qzz 〉
(A3)
3 (n 0 λ )−1⋅ab . 2
where С′= 6 × Averaging the operator Qzz by the ground state/ LSJm> for the Tb3+ RE-ion, one can obtain [7]:
β0 = C′⋅(L // n1// L)2 ∑ β′m (J )
(A4)
m
In this approximation A4 can be rewritten for β′m as follows (see equation (9) in Ref. [7]):
β′m (J ) =
ρm =
L S J⎫ (2J + 1) ⋅(−1) L + S + J ⎧ 2 J L⎬ ⎨ ⎭ ⎩
∑ m
[3m2 − J (J + 1)] (2J − 1) J (J + 1)(2J + 3)
ρm
(A5)
exp(−mgμB H / kB T ) ∑m exp(−mgμB H / kB T )
where g = 1.5 is the Lande factor of the ground 7F6 multiplet of the Tb3+ ion. The value of β0 is a phase shift caused by MLB in the absence of Van Vleck's “mixing” of the neighboring multiplets wavefunctions by the external magnetic field H [7]. Note that in the above-mentioned work [7] the influence of the magnetic “mixing” of the RE-ion multiplets (particularly, for Sm3+ ion) on the formation mechanisms of quadratic magneto-optical effects in RE-garnets is considered in detail. Since the parameter x = mgμB H / kB T at high temperatures (and small magnetic fields) satisfies the condition x < < 1, the expression for the Boltzmann population can be decomposed into a Taylor series up to members of the second order of smallness. After the summation in expression (A5) by index m is made, it is easy to find that:
∑m β′m (J ) =
= 6.493 × 10−3 ⎡ ⎣ Since terms)
1 2
m =+J
∑ m =−J
[3m2 − J (J + 1)]
L S J ⎫ m =+J (2J + 1) ⋅⎧ ∑ 2 J L ⎬ m =−J ⎨ ⎭ ⎩
m =+J ∑m =−J
J (J + 1) gμB H 2 ⎤ kB T ⎦
[3m2
(2J − 1) J (J + 1)(2J + 3)
⎡1 − ⎣
mgμB H kB T
+
1 2
(
mgμB H 2 kB T
)
+ ...⎤= ⎦
μ H 2
= 25.77 ⎡ kB T ⎤ ⎣ B ⎦
(A6)
− J (J + 1)] = J (J + 1)(2J + 1) − J (J + 1)(2J + 1) =
m =+J 0∑m =−J
[3m2
− J (J +
mgμ H 1)] k BT B
= 0 (because of odd character of the sum
mgμB H ⎞2 J (J + 1) gμB H ⎤2 [3m2 − J (J + 1)] ⎛ = 1.755 ⎡ ⎢ ⎥ kB T ⎝ kB T ⎠ ⎣ ⎦ ⎜
⎟
Using expression (A4) and substituting formulas (A1) and (A5), we can calculate the expression for the phase shift caused by the MLB in TbGG and write it in the approximation of the “free RE-ion” as follows:
β0 (Tb3 +) = 15.146
μ H 2 N (n¯ 2 + 2)2 ωω (erfd )2 ⎛ B ⎞ ⎡ 2 0 2 ⎤ ⎥ 9nπc ¯ ℏ ⎝ kB T ⎠ ⎢ ⎣ (ω0 − ω ) ⎦ ⎜
3+
where N is the number of the Tb
⎟
(A7)
ions per unit of volume, n¯ is the refractive index of garnet, μ B is the Bohr magneton.
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