Faraday effect and magnetic susceptibility analyses in TBALO3

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Faraday effect and magnetic susceptibility analyses in TbAlO₃

Uygun V. Valiev, Abdulla A. Uzokov, and Sharof A. Rakhimov

Faculty of Physics, National University of Uzbekistan, Tashkent, 100174, Uzbekistan

John B. Gruber, Kelly L. Nash, and Dhiraj K. Sardar

Department of Physics and Astronomy, University of Texas at San Antonio, San Antonio, Texas 78249-0697, USA

Gary W. Burdick

Department of Physics, Andrews University, Berrien Springs, Michigan, 49104, USA

It is well-known that a detailed study of the magnetic and magnetooptical phenomena, sensitive to the features of the electronic structure of Tb³⁺ in the low-symmetry (C_S) sites in the orthoaluminate structure of TbAlO₃, is able to provide better understanding of the role that the Van-Vleck mechanism for "admixing" excited states with the thermally-populated ground state of magnetoactive ion plays in this crystal [1-2]. In this work the analysis of the Faraday effect and magnetic susceptibility of TbAlO₃ was made in accordance with suggestion that Van-Vleck mechanism plays a noticeable role in the formation of the magnetooptical [3] and magnetic [1-2, 4] properties of a Tb³⁺ ion in the YAlO₃ host.

Magnetic susceptibility of the TbAlO₃ Since Tb³⁺(4f⁸) is a non-Kramers ion, each ^2S+1L_J multiplet manifold splits into 2J+1 non-degenerate Stark levels (singlets), when the ion enters the low-symmetry (C_S) cation sites in the orthorhombic lattice (space group Pbmn) [1-2]. As a result, the magnetization of the non-Kramers RE-sublattice in the orthoaluminate structure may arise only when the effect of “mixing” in an external magnetic field H (Van Vleck mechanism) is taken into account [2]. Earlier studies [1-2], have shown that Tb³⁺ in TbAlO₃ can be treated as an “Ising” ion with its “Ising” axis taken as the anisotropy axis lying in the a-b plane of the orthorhombic crystal at an angle of ₀ = 36⁰ to the a-axis of crystal. We then choose the “Ising” axis as the z-axis of the local coordinate system of the Tb³⁺ ion (located at one of the non-equivalent sites) so that the y-axis will be parallel to the c-axis of the orthorhombic crystal. It is important to note that the wave functions of the ground quasi-doublet, formed by two nearly degenerate Stark singlets that have different irreducible representations (irreps) of the C_S group (A and B), can be approximated very well by a linear combination of “pure” |J,±M_J> states of the type |6, ±6> at the same orientation of the local axes (the so-called “Ising” orientation) [1-2,4]. The wave functions of the first excited states at 163 cm^-1 and 199 cm^-1 have different symmetries A and B as well and can be represented by a linear combinations of |6,±5> states [4]. The eigenvectors representing these states are listed in Table V of ref. [4] and were used to calculate the molar susceptibility for TbAlO₃ in the ab - plane of the rhombic crystal as:

(1)

(1a)

(1b)

where N_A is Avogadro's number; n m; E_n and E_m are the energies of the “mixing” states; /n>, /m> and /i>, /k> are the real wave functions of the non-degenerate Stark sublevels; _n is the Boltzmann population; is the contribution to a magnetic susceptibility due to differences in the population of the Stark levels shown in the inset (longitudinal susceptibility), and is the Van-Vleck contribution to a magnetic susceptibility, (having an isotropic nature in comparison with the magnetic susceptibilities along -a and -b axes of rhombic crystal) [1,4]. Using the wave functions and the Stark levels from the far right column of TableV in [4], we calculate expressions for the longitudinaland transverse magnetic susceptibility to for the temperature range between 10 K and 300 K, which have the following form:

= (2a)

= (2b)

where, ₀ is the Boltzmann population of the Stark levels of the ground state quasi-doublet, and _1-7 are the Boltzmann populations of Stark levels at energies, 163, 199, 207, 257, 295, 343, and 365 (all in cm^-1), respectively [4]; _p 5K is the “paramagnetic” Curie temperature [2]; and g₀ = 1.5 is the Lande factor of the ⁷F₆ ground-state multiplet. In determining values for, we consider that the Stark singlets are non-spin states, so that the Van-Vleck's magnetic moment can be oriented only along the direction of an external magnetic field. The inverse magnetic susceptibility(where M is the molecular weight of TbAlO₃) as a function of temperature based on the calculated susceptibility that is obtained from Eqs. (1) and (2) is in good agreement with the experimental values of the magnetic susceptibilityinvestigated in the temperature range between 80 K and 300 K. The agreement confirms that the group-theoretical irrep labels and the energy for the Stark levels have been properly assigned for the energy Stark levels in the ⁷F₆ ground-state manifold [4]. Further confirmation of the Stark splitting assignments given to the ⁷F₆ manifold can be obtained by analyzing the Faraday rotation observed in TbAlO₃ in an external magnetic field H oriented along the crystallographic direction [110].

Faraday Rotation of the TbAlO₃

Since TbAlO₃, in the optical sense, are biaxial crystal we run into problems when investigating linear magnetooptical effects such as the Faraday effect (FE), because of a “background” coming from a large natural birefringence (n 10^-2) [5]. According to the phenomenological theory for the Faraday effect (FE) developed in [5], the oscillating character of rotation angle of the major axis of the polarization ellipse is associated with an increase in the natural birefringence n as the temperature decreases from 300 K to 85 K (see Fig.1).

Figure 1

Temperature dependences of the rotation angle measured along [110] axis in TbAlO₃ at the = 506 nm. The inset presents the dependence of the Verdet constant V of TbAlO₃ measured along [110] axis from longitudinal magnetic susceptibility.

As the temperature decreases, the amplitude of the angle oscillations shown in Fig.1 becomes larger, resulting in an increase in the Verdet constant V at lower temperatures. This is noted in the insert in Fig.1, where the Verdet constant is plotted as a function of the longitudinal susceptibility which is given in Eq.2a. It is well seen that the experimental data points are connected very well with the results from our calculations.

To interpret the experimental data plotted in Fig.1 that reflect the contribution of the Van Vleck “mixing” in the FR, we calculated the temperature-dependent Faraday effect using the wave functions and irreducible representations (irreps) reported in [4]. We first recognize that the FR along the [110] axis depends on the opposite symmetry of the two lowest-energy Stark levels in the quasi-doublet (a, b) and the excited level j. The “paramagnetic” term of the FR along the z-axis (“Ising axis”) of the local coordinate system is proportional to the product of the matrix elements of the electrodipole (ED) transitions between the a, b states of the ground-state quasi-doublet and the excited state j, and the magnetic dipole transitions between the Stark levels within the quasi-doublet. For one of the two non-equivalent Tb³⁺ sites, the Faraday rotation angle for ED transitions at frequencies far from the resonance frequency ₀ is given as [3,6]:

_F^C= (3)

where N is the number of Tb³⁺ ions per volume (cm³),

n is the index of refraction, d₀ is the ground-state degeneracy,

and the x - and y- components of the dipole moments of Tb³⁺ are transformed according to the irreps.

B, A and B of the Cs group, respectively. The excited state j is associated with the 4f^(n-1)5d configuration of Tb³⁺(4f⁸), where the optical transition has a average frequency ₀. Based on group theory arguments alone, the existence of the opposite symmetry of states within the ground-state quasi-doublet is confirmed from the analysis of the magnetic susceptibility [1-2] and the crystal-field splitting calculations and wave functions obtained from [4]. By applying the Wigner-Ekkart theorem and selection rules for the spin and parity allowed ED 4f 5d transition between the ground (L₀S₀J₀) multiplet of the 4f⁽ⁿ⁾ configuration and the (L = L₀ - 1, S₀) term of the excited 4f^(n-1)5d configuration of Tb³⁺, as well as by neglecting the crystal-field splitting in the summation over excited states by using the Judd-Ofelt approximation, Eq. (3) can be expressed as:

_F^C = KM₀ (4)

(5)

faraday magnetic optical susceptible

where, g₀ = 1,5 for Tb³⁺; is the genealogical coefficient (or coefficient of fractional parentage);

is the 6j - symbol [7];

L = 2 - orbital moment of excited term ⁷D belonging to the “mixed” 4f^(n-1)5d;

L₁ = 0 - orbital moment of 4f⁽⁷⁾ “core” of Tb³⁺;

L₀ = 3 and S₀ = 3 - orbital and spin moments of the ground state 4f⁽⁸⁾ of Tb³⁺, accordingly.

Note, that the “constant K” in Eq. (4) for the Tb³⁺ ion can be approximated by examining its form in Eqn. (5). For example, <r> = (4f/r/5d) is the radial integral for r between the 4f and 5d states, values of which can be modeled (see also [8]). The magnetic moment of the RE-sublattice in TbAlO₃ at low T in Eq.(4) is associated with the difference in the populations of the Stark levels of the ground state quasi-doublet [1]

From Eq. (3) we see that the actual (and unknown) wavefunction of the Stark singlet - j belonging to the mixed excited 4f⁽⁷⁾5d configuration of Tb³⁺ in TbAlO₃ to which the allowed optical transition takes place, is transformed according to the irreducible representation B and contains the ⁷D₅ state with large enough weight (1):⁷D₅/B>=. A phenomenological “K” can also be assigned so that a smooth curve can pass through the data points plotted in Fig. 1. The numerical value for K in Eq.(5) is 0,943 (cm^-1Oe^-1) and the empirical value is 0,87 (cm^-1Oe^-1). The difference reflects the uncertainty and assumptions made in calculating the given magnetooptical constant.

The temperature-independent (or weakly temperature-dependent) contribution _F^B to the FR (i.e. B - term FR [3,6]) along the b-axis (or x-axis of the local coordinate system) is connected by mixing the opposite symmetry Stark levels - a and - d of the ground state and the first excited state having symmetry A and B, respectively. This FR contribution is given as:

_F^B = (6)

where, is the energy separation between the states that are mixed.

By taking into account Van Vleck “mixing” and selection rules for the ED transitions, formula (6) can be rewritten as:

_F^B = KM_VV. (7)

where, M_VV is the Van-Vleck contribution at low T [1] to the resulting magnetization of the RE-sublattice; coefficient K was defined earlier in Eq.(5). It must be emphasized that according to (4) and (7) the FR in TbAlO₃ depends on resulting magnetization of RE-sublattice TbAlO₃:

M = M₀ + M_VV,

and consequently, in a general case, we have the following expression for the Verdet constant V of TbAlO₃:

V = V^C + V^B = K( + ) (8)

where the longitudinal and transverse (Van-Vleck) susceptibilities are defined earlier by Eqs.(2a) and (2b).

However, selection rules involving the matrix elements between the - a and - d states and the excited states in the mixed 4f⁽⁷⁾5d configuration are zero for all allowed (spin and parity) optical transitions. As a consequence of this, the temperature dependence of the Verdet V constant of TbAlO₃ along the [110] plane will be determined only by the first term (8), i.e. V_[110] , which agrees with the results of magnetooptical experiments presented in Fig.1 in the temperature interval 85 ч 300K.

References:

[1] Zvezdin A.K., Matveev V.M., Mukhin A.A., and Popov A.I. Rare Earth Ions in Magnetic Ordered Crystals (Nauka, Moscow, 1985).

[2] Holmes L., Sherwood R., and Van Vitert L. G.J. Appl. Phys. 1968 39 1373

[3] Valiev U.V., Zvezdin A. K., Krinchik G.S., Levitin R.Z., Mukimov K.M., and Popov A.I. Sov. Phys. JETP 1983, 58(1), 181

[4] Gruber J.B., Sardar D.K., Nash K.L., Yow R.M., Valiev U.V., Uzokov A. A., and Burdick G.W. Journ. of Lumin. 2007

[5] Valiev U.V., Klochkov A.A., Lukina M. M., and Turganov M.M. Opt. Spectros. 1987, 63, 319

[6] Stephens P.J. Advan. Chem. Phys. 1976, 35, 197.

[7] Morrison C.A. Angular Momentum Theory Applied to Interactions in Solids (Springer, New York, 1988).

[8] Judd B.R. Phys. Rev. 1962, 127, 750.

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