Spin-Hamiltonian Parameters of Gd Ion in the Room Temperature Tetragonal Phase of BaTiO

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Spin-Hamiltonian Parameters of Gd Ion in the Room Temperature Tetragonal Phase of BaTiO

₃

Wei-Qing Yang^a,band Wen-Chen Zheng^c

a Key Laboratory of Advanced Technologies of Materials (Ministry of Education), School of Materials Science and Engineering, Southwest Jiaotong University, Chengdu 610031, China

b State Key Laboratory of Electronic Thin films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China

c Department of Material Science, Sichuan University, Chengdu, China Reprint requests to W.-Q. Y.; E-mail:wqyang@home.swjtu.edu.cn Z. Naturforsch.69a, 606 – 610 (2014) / DOI: 10.5560/ZNA.2014-0057

Received April 24, 2014 / revised July 10, 2014 / published online August 27, 2014

The spin-Hamiltonian parameters (gfactorsg_k,g_⊥, and zero-field splittingsb⁰₂,b⁰₄,b⁴₄,b⁰₆,b⁴₆) of the 4f⁷Gd³⁺ion in the tetragonal phase of a BaTiO3crystal are calculated through the diagonalization (of energy matrix) method based on the one-electron crystal field mechanism. In the calculations, the crystal field parameters are estimated from the superposition model with the structural data of the studied crystal. It is found that by using three adjustable intrinsic parameters ¯A_k(R0) (k=2, 4, 6) in the superposition model, the seven calculated spin-Hamiltonian parameters are in good agreement with the experimental values, suggesting that the diagonalization method based on one-electron crystal field mechanism is effective in the studies of spin-Hamiltonian parameters for 4f⁷ions in crystals.

Key words:Electron Paramagnetic Resonance; Spin-Hamiltonian Parameters; Crystal Field Theory;

Gd³⁺; BaTiO3.

1. Introduction

Barium titanate (BaTiO₃) doped with rare earth (RE) ions have received attention for applications as the dielectric of multilayer ceramic capacitors, the positive temperature coefficient of resistance thermis- tors, and luminescence materials [1–4]. Recently, the anomalous photoferromechanical effect and negative photoconducivity were reported in BaTiO₃:Gd³⁺ [3].

So, a variety of techniques were applied to study the properties of RE-doped BaTiO3 crystals [1–9]. May be because the electron paramagnetic resonance (EPR) spectra of Gd³⁺-doped materials can be obtained even at room temperature, the EPR spectra of Gd³⁺ ions in the room temperature tetragonal phase of BaTiO₃ were measured decades ago [9]. The measurements suggested that a Gd³⁺ ion in the tetragonal phase of BaTiO₃occupies the 12-coordinated Ba²⁺site, and no local charge compensation in the nearby surroundings of the Gd³⁺ ion is found because when BaTiO₃ be- comes the high temperature (T ≈425 K) cubic phase, this Gd³⁺center is also changed to a cubic one [9]. The site symmetry of the Gd³⁺ ion in tetragonal BaTiO3

is C_4v, and the spin-Hamiltonian parameters (g fac- torsg_k,g_⊥, and zero-field splittingsb⁰₂,b⁰₄,b⁴₄,b⁰₆,b⁴₆) of this tetragonal Gd³⁺ center were determined [9].

However, until now no theoretical calculations have been done for these spin-Hamiltonian parameters, per- haps because the 4f⁷ ions are S-state ions. Due to the lack of orbital angular momentum, the micro- scopic mechanisms of spin-Hamiltonian parameters are complex [10–14]. Besides the important and conventional one-electron crystal field mechanism, other crystal field mechanisms, such as correlation (e.g., spin-correction [11,14,15]), relativistic and quadratic crystal field mechanisms, can contribute to the spin- Hamiltonian parameters [10–14]. These other crystal field mechanisms are partly canceled [10–14], and the conventional one-electron crystal field mechanism can interpret rationally the spin-Hamiltonian parameters for the Gd³⁺ ion in crystals when the crystal field parameters are estimated from the superposition model rather than the electrostatic model [16–19]. For example, by using the crystal field parameters estimated from the superposition model with the structural data obtained from the analyses of superhyperfine in-

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W.-Q. Yang and W.-C. Zheng·Spin-Hamiltonian Parameters of Gd Ion in Tetragonal BaTiO3 607 teraction constants [20], the spin-Hamiltonian param-

eters of the tetragonal Gd_M³⁺– F_i⁻ centers in CaF₂ and SrF₂crystals are reasonably explained through the one-electron crystal field mechanism [17]. So, in this paper, we calculate the spin-Hamiltonian parameters of Gd³⁺in the tetragonal BaTiO₃crystal through the diagonalization (of energy matrix) method based on the conventional one-electron crystal field mechanism.

The results are discussed.

2. Calculation

The free Gd³⁺ion has the ground state⁸S_7/2. It occupies the 12-coordinated Ba²⁺ site in the tetragonal BaTiO₃[1,2,9]. The tetragonal crystal field acting on this site can split the ground multiplet⁸S_7/2into four Kramers doublets, and in an EPR experiment the external magnetic field splits further these doublets into eight singlets withJ=−7/2,−5/2,−3/2,−1/2, 1/2, 3/2, 5/2, and 7/2, respectively. The EPR spectra of tetragonal Gd³⁺centers can be described by the effective spin Hamiltonian [21]

H_s=g_kβB_zS_z+g⊥β(B_xS_x+B_yS_y) +1

3b⁰₂O⁰₂+ 1

60(b⁰₄O⁰₄+b⁴₄O⁴₄) + 1

1260(b⁰₆O⁰₆+b⁴₆O⁴₆)

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in whichOⁿ_m represent the Stevens operators [21,22].

The spin-Hamiltonian parameters in (1) are connected with the angular dependences of EPR transitions among the above eight singlets (e.g., for BaTiO₃:Gd³⁺

considered, see Fig. 3 in [9]). The correlations among the spin-Hamiltonian parameters and the energy level differences concerning EPR transitions∆E(J→J−1) are derived in [18,19] where these EPR transitions are obtained from the second-order perturbation calculations in [21], and the results are as follows [18,19]:

When the external magnetic fieldBis parallel to the C₄- (orz-)axis, the energy differences∆E_j(k)concerning the EPR transitions∆E(J→J−1)are

∆E₁(k) =∆E 3

2 →1 2

−∆E

−1 2 → −3

2

=2(2b⁰₂−12b⁰₄+14b⁰₆),

∆E2(k) =∆E 5

2 →3 2

→∆E

−3 2 → −5

2

=2(4b⁰₂−10b⁰₄−14b⁰₆),

∆E3(k) =∆E 7

2 →5 2

−∆E

−5 2→ −7

2

=2(6b⁰₂+20b⁰₄+6b⁰₆),

∆E 1

2→ −1 2

≈g_kβB_z. (2) When the external magnetic fieldB is perpendicular to the z-axis and parallel to the x-axis, the energy differences ∆Ej (⊥) concerning the EPR transitions

∆E(J→J−1)are

∆E₁(⊥) =∆E 3

2→1 2

−∆E

−1 2→ −3

2

=2b⁰₂+9b⁰₄−35

4 b⁰₆+3b⁴₄+7 4b⁴₆,

∆E₂(⊥) =∆E 5

2→3 2

−∆E

−3 2→ −5

2

=4b⁰₂+15 2 b⁰₄−35

4 b⁰₆+5 2b⁴₄−7

4b⁴₆,

∆E₃(⊥) =∆E 7

2→5 2

−∆E

−5 2→ −7

2

=6b⁰₂−15b⁰₄+15

4 b⁰₆−5b⁴₄+3 4b⁴₆,

∆E 1

2→ −1 2

≈g⊥βBx. (3) The Hamiltonian of the Gd³⁺ion in a tetragonal crystal field and under an external magnetic field is written as H=H_f+H_CF+H_ze, (4) where the three terms are, respectively, free-ion Hamil- tonian, crystal field interaction (based on one-electron crystal field mechanism), and Zeeman (or magnetic) interaction terms. They can be expressed as [22,23]

H_f=E_AVE+

∑

k=2,4,6

F^kf_k+ζ4fA_SO+αL(L+1)

+βG(G₂) +γG(R₇) +

k=2,4,6

∑

t=2,3,4,6,7,8

t_iT_k +

∑

k=0,2,4

m_kM^k+

∑

k=2,4,6

p_kP^k,

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HCF=B⁰₂C₂⁰+B⁰₄C₄⁰+B⁴₄(C₄⁴+C⁻⁴₄ )

+B⁰₆C₆⁰+B⁴₆(C⁴₆+C₆⁻⁴), (6) H_Ze=g_JβJ·B, (7) in which B^q_k are the crystal field parameters and the remaining notations take the standard meanings [22, 23].

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608 W.-Q. Yang and W.-C. Zheng·Spin-Hamiltonian Parameters of Gd Ion in Tetragonal BaTiO3

The splitting of the ground multiplet⁸S_7/2into eight singlets is caused by the interactions of it with the excited multiplets via crystal field and external magnetic field. So, the energy levels of the eight singlets are related to the eigenvalues of the energy matrix corresponding to the Hamiltonian in (4). Consider- ing that the splitting of the ground multiplet is due mainly to the interaction of it with the low-lying excited multiplets with the same value of J(=7/2)as the ground multiplet [10,14,24,25], as an approximation, we found the 56×56 dimension energy matrix of the Hamiltonian in (4) consisting of the ground multiplet⁸S_7/2and the low-lying excited multiplets⁶L_7/2 (L=P,D,F,G,H,I) with the aid of irreducible ten- sor and/or equivalent operator method. Diagonalizing the energy matrix, the energy levels E(J)(the eigenvalues of the energy matrix) of the eight singlets from the ground state ⁸S_7/2and hence the EPR transitions

∆E(J→J−1)between these energy levels can be obtained. Thus, the energy differences ∆E_j(k) and∆E_j (⊥) as well as the spin-Hamiltonian parameters can be calculated from (2) and (3).

In the energy matrix, the free-ion parameters in (4) for the Gd³⁺ion are taken as the mean values acquired from Gd³⁺ions in lots of crystals [23] (see Tab.1) because they only change slightly from crystal to crystal.

The crystal field parameters B^q_k of rare earth ions in crystals are frequently calculated from the superposition model [12,13]. The model proposes that the crystal field parametersB^q_k come from a sum of contribu- tions due to all ligands [12,13], namely,.

B^q_k=

∑

i

A_k(R₀) R₀

Ri

t_k

K_kq(θ_i,ϕi)/α_k0, (8) where the values ofα_k0(k=2, 4, 6) are shown in [13];

t_k are the power-law exponents. For rare earth ions in crystals, we taket₂≈5,t₄≈6,t₆≈10 [26,27]. ¯A_k(R₀) are the intrinsic parameters, andR₀is the reference distance.K_kq(θ_i,φ_i)are the coordination factors depend- ing on the sites of the ligands. Ri (the metal–ligand distance), θi (the angle between R_i and z axis), and φ_i (the azimuthal angle) are the structural data of the

Table 1. Mean values of free-ion parameters (in cm⁻¹) of the Gd³⁺ion [20].

F² F⁴ F⁶ α β γ T² T³ T⁴ T⁶

85 300 60 517 44 731 18.95 −620 1658 308 43 51 −298

T⁷ T⁸ ζ4f M⁰ M² M⁴ P² P⁴ P⁶

338 335 1504 2.99 1.67 1.14 542 407 271

studied system. The 12 O²⁻ligands in the (GdO₁₂)²¹⁻

clusters in the tetragonal phase of BaTiO₃ can be di- vided into three groups. The four O²⁻ligands in each group have the same distanceR_i and angleθi, but the azimuthal angles of four O²⁻ligands in groups 1 and 2 are 0^◦, 90^◦, 180^◦, 270^◦and those in group 3 are 45^◦, 135^◦, 225^◦, 315^◦[28]. Thus, from (8), for the tetragonal (GdO12)²¹⁻clusters in BaTiO3, we have

B⁰₂=4A₂(R₀)

3

∑

i=1

R₀ R_i

t₂

(3 cos²θ_i−1),

B⁰₄=4A₄(R₀)

3 i=1

∑

R₀ Ri

t4

(35 cos⁴θi−30 cos²θi+3),

B⁴₄=2√

70A₄(R₀) ( 2

∑

i=1

"

R₀ R_i

t₄

sin⁴θ_i

#

− R₀

R3

t4

sin⁴θ3

) ,

B⁰₆=4A₆(R₀)

3 i=1

∑

R₀ Ri

t6

(231 cos⁶θi−315 cos⁴θi

+105 cos²θi−5), B⁴₆=6√

14A₆(R₀)

· ( 2

i=1

∑

"

R₀ R_i

t6

(11 cos²θi−1)sin⁴θi

#

− R0

R₃ t₆

(11 cos²θ₃−1)sin⁴θ₃ )

. (9)

The structural dataR^h₁≈2.795 Å,θ₁^h≈45.61^◦,R^h₂≈ 2.883 Å,θ₂^h≈43.87^◦,R^h₃≈2.842 Å, andθ₃^h≈87.97^◦ in the host BaTiO₃crystal were obtained from X-ray and neutron diffraction [28]. It is generally agreed that the local structural data of an impurity center in crystals may be unlike the corresponding structural data in the host crystal because of the size and/or charge mis- match [29–31]. No local structural data of the Gd³⁺

impurity center in BaTiO₃were reported. We estimate rationally them as follows: No local charge compensation was found in the nearby surroundings of Gd³⁺in

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W.-Q. Yang and W.-C. Zheng·Spin-Hamiltonian Parameters of Gd Ion in Tetragonal BaTiO3 609 Table 2. Spin-Hamiltonian parameters (gfactorsg_k,g⊥, and zero-field splittingsbⁿ_m,bⁿ_mare given in 10⁻⁴cm⁻¹) for the Gd³⁺

ion in the tetragonal phase of BaTiO₃.

g_k≈g_⊥ b⁰₂ b⁰₄ b⁴₄ b⁰₆ b⁴₆

Calc. 1.993 −292.1 3.8 −1.6 0.9 0.01

Expt. [9] 1.995(3) −293.6(10) 4(1) −2(1) 1.6(10) ∼0

BaTiO₃[9], thus the changes of bonding anglesθi(i.e., angular distortions) in the (GdO12)²¹⁻clusters may be small because of the lack of the perturbation of charge compensator. In order to decrease the number of adjustable parameters, as an approximation, we neglect the small changes of bonding angles θi and consider only the changes of the metal–ligand distancesR_idue to the difference between the ionic radiusr_iof the impurity and the radius r_hof the replaced host ion. An approximate formula R≈R_h+¹₂(r_i−r_h)[30,31] is often used to estimate reasonably the metal–ligand dis- tanceRin an impurity center in crystals. For Gd³⁺in BaTiO₃under study, from the above host metal–ligand distances R^h_i, r_i(Gd³⁺)≈0.938 Å and r_h(Ba²⁺)≈ 1.34 Å [32], we getR1≈2.594 Å,R2≈2.682 Å, and R₃≈2.641 Å. The reference distance is taken asR₀≈ R¯≈2.639 Å. Thus, in the energy matrix, there are only three parameters ¯A_k(R₀)left as adjustable parameters.

By matching the calculated spin-Hamiltonian parameters by diagonalizing the energy matrix with the experimental values, we obtain for the tetragonal (GdO₁₂)²¹⁻

clusters in BaTiO₃

A¯2(R₀)≈452 cm⁻¹, A¯₄(R₀)≈156 cm⁻¹, A¯₆(R₀)≈10 cm⁻¹.

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The calculated spin-Hamiltonian parameters are compared with the experimental values in Table2.

3. Discussion

In the superposition model, the intrinsic parameters ¯A_k(R₀)are treated as adjustable parameters. There may be some errors in the above intrinsic parameters ¯A_k(R₀) in BaTiO₃:Gd³⁺ because in the superposition model calculations the small angular distortions of (GdO₁₂)²¹⁻ clusters in BaTiO₃ are ne- glected. Even so, these intrinsic parameters can be regarded as suitable because of the following rea- sons: (i) From a great number of studies for the crystal field parameters in optical and EPR spectra of

RE³⁺-doped crystals using the superposition model, a rule of ¯A2(R₀)>A¯4(R₀)>A¯₆(R₀)was found [12, 13,16–19,26,27,33–38]. The values of ¯A_k(R₀)obtained from the above calculation for BaTiO₃:Gd³⁺are consistent with the rule. (ii) Many studies also suggested that the intrinsic parameters ¯A_k(R₀)for the same RE³⁺–X combination in similar clusters and crystals may be near because of similar immediate environ- ments of the RE³⁺ions [26,27,33–37]. The intrinsic parameters ¯A₂(R₀)≈470 cm⁻¹, ¯A₄(R₀)≈140 cm⁻¹, and ¯A₆(R₀)≈ 8 cm⁻¹ (with similar distance R₀≈ 2.758 Å) for the tetragonal (GdO₁₂)²¹⁻clusters in the tetragonal SrTiO₃ were reported [19]. It can be seen that the corresponding parameters ¯A_k(R₀) of similar (GdO₁₂)²¹⁻clusters in both BaTiO₃and SrTiO₃crystals are close to each other and so the parameters A¯_k(R₀) in BaTiO₃:Gd³⁺ are proper. Thus, by using the diagonalization (of energy matrix) method based on the one-electron crystal field mechanism with three proper adjustable parameters ¯A_k(R₀), the calculated spin-Hamiltonian parametersg_k,g⊥,b⁰₂,b⁰₄,b⁴₄,b⁰₆, and b⁴₆for Gd³⁺in tetragonal BaTiO₃crystal are in reason- able agreement with the observed values (see Tab.2).

4. Conclusion

Seven spin-Hamiltonian parameters of the Gd³⁺

ion in the tetragonal BaTiO₃ crystal can be reasonably explained from the diagonalization (of energy matrix) method based on the one-electron crystal field mechanism, in which the crystal field parameters are estimated from the superposition model. It appears that this method is effective to the studies of spin- Hamiltonian parameters for 4f⁷ (e.g., Gd³⁺) ions in crystals.

Acknowledgement

This work is supported by the National Natural Sci- ence Foundation of China (No. 51202023), China Post- doctoral Science Foundation (No. 2013T60845 and 2012M511917), and the Fundamental Research Funds for the Central Universities (A0920502051408-10).

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610 W.-Q. Yang and W.-C. Zheng·Spin-Hamiltonian Parameters of Gd Ion in Tetragonal BaTiO3

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