Analyses of the crystal field energy levels for Cr3+-doped Gd3Sc2Ga3O12 garnet crystal

Optik 152 (2018) 69–72

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Original research article

Analyses of the crystal field energy levels for Cr3+ -doped Gd3 Sc2 Ga3 O12 garnet crystal Bo-Wei Chen a , Yang Mei a,∗ , Wen-Chen Zheng b a b

School of Mathematics & Physics, Mianyang Teachers’ College, Mianyang 621000, PR China Department of Material Science, Sichuan University, Chengdu 610064, PR China

a r t i c l e

i n f o

Article history: Received 12 June 2017 Accepted 25 September 2017 Keywords: Optical spectrum Crystal- and ligand-field theory Defect structure Gd3 Sc2 Ga3 O12 Cr3+

a b s t r a c t The crystal field energy levels of Cr3+ -doped Gd3 Sc2 Ga3 O12 (GSGG) garnet crystals are calculated from the complete diagonalization (of energy matrix) method based on the twospin-orbit-parameter model, where the contributions due to both the spin-orbit parameter of central dn ion and that of ligand ions are contained. The calculated results indicate that the seven crystal field energy levels available in experiments are rationally explained by this method with only three adjustable parameters. On account of the calculations, the defect structure (specifically, the angular distortion) of the trigonal Cr3+ impurity center in GSGG crystals is also evaluated. The results are discussed. © 2017 Published by Elsevier GmbH.

1. Introduction Gd3 Sc2 Ga3 O12 (GSGG) is one of the numbers of garnet family. It can be grown easily large size without inhomogeneous central cores [1] in comparison with the well-known Y3 Al5 O12 (YAG) garnet crystal. Importantly, GSGG doped with Cr3+ ions is a good tunable laser crystal at room temperature in the near infrared spectral range [2–4]. So, considerable internets are focus on the spectroscopic properties of GSGG: Cr3+ crystal because they can provide information on the crystal field energy levels, substitutional site and defect structure of the active ion center in crystals. The optical spectra, including the ground-state absorption and the excited-state absorption spectra, for GSGG: Cr3+ crystals were measured experimentally [3–7]. The measurements found that Cr3+ ions in GSGG crystals occupy the trigonal octahedral Sc3+ site and seven crystal field energy levels were observed in the crystals [3–7]. Theoretically, unfortunately, the calculations or explanations for all these crystal field energy levels, particularly, those connected with the defect structure of Cr3+ active center in GSGG crystals, have not been made. It is known that the crystal field energy levels of a dn ion in crystals can be computed by the complete diagonalization (of energy matrix) method because they correspond with the eigenvalues of the energy matrix [8,9]. Since the observed crystal field energy levels for GSGG: Cr3+ crystals consist of the splitting of first excited state 2 E which is closely related to the spin-orbit parameter [10], the Hamiltonian of energy matrix should contain the spin-orbit interaction. In the classical crystal field theory, only the influences of spin-orbit parameter of central dn ion (i. e., one-spin-orbit-parameter model) are taken into account [9]. For more completeness, hence we investigate the crystal field energy levels of GSGG: Cr3+ crystals in virtue of the complete diagonalization (of energy matrix) method founded on the two-spin-orbit-parameter model, where besides the above influences of the spin-orbit parameter of dn ion, those of ligand ions via covalence effect

∗ Corresponding author. E-mail address: [email protected] (Y. Mei). https://doi.org/10.1016/j.ijleo.2017.09.097 0030-4026/© 2017 Published by Elsevier GmbH.

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are also considered [11,12]. The defect structure (in particular, the angular distortion) of trigonal octahedral Cr3+ center in GSGG: Cr3+ crystals is also evaluated on the basis of the calculations. The results are discussed. 2. Calculation The Hamiltonian of d3 ion in a trigonal octahedral system based on the two-spin-orbit-parameter model is constituted by the Coulomb, crystal field and spin-orbit interaction terms, i. e., [11,12] H = H Coul. (B, C) + H CF (B20 ,B40 ,B43 ) + H SO (,  )

(1) 

d3

where B and C are Racah parameters of ions in crystals. Bkl are the crystal field parameters,  and are two spin-orbit parameters required in the two-spin-orbit-parameter model [11,12]. The complete energy matrix of this Hamiltonian is 120 × 120 dimensions and is constructed in terms of the strong field functions [13]. In the energy matrix, the two spin-orbit parameters ,   can be computed by the equations [11,12]  = Nt (d0 + 1

1 2 0   ) 2 t p

  = (Nt Ne ) 2 (d0 −

1 t e p0 ) 2

(2)

where  d 00 and  p 00 are the spin-orbit parameters of dn and ligand ions in the free state. For the GSGG: Cr3+ crystals under consideration, one can find  d 00 (Cr3+ ) ≈ 273 cm−1 [14] and  p 00 (O2− ) ≈ 150 cm−1 [15]. N ( = t or e) and  are the molecular orbital coefficients in the one-electron basis functions based on the two-spin-orbit-parameter model [11,12]. They are related and calculated by the normalization relations [11,12]





N 1 − 2 Sdp () + 2 = 1 and the approximate correlations



(3)



2 f = B/B0 = C/C0 = N2 1 − 2 Sdp () + 2 Sdp ()

(4)

where B0 and C0 are the Racah parameters of free d3 ions. Here, we have B0 ≈ 1030 cm−1 and C0 ≈ 3850 cm−1 [14] for free Cr3+ ion. Sdp () are the group overlap integrals which can be computed from the Slater-type self-consistent field (SCF) functions [16,17] with the metal-ligand distance R in the studied system. It is agreed that the metal-ligand distance R in an impurity center in crystals may be unlike the corresponding distance Rh in the host crystal owing to the difference between the ionic radii ri of impurity and rh of the replaced host ion [18]. Here we employ an approximate formula R ≈ Rh + 12 (ri − rh ) [18] to estimate the distance R. In GSGG: Cr3+ crystal, we have Rh ≈ 2.088 Å [19], ri (Cr3+ ) ≈ 0.755 Å and rh (Sc3+ ) ≈ 0.885 Å [20], thus, we get R ≈ 2.023 Å and hence Sdp (e) ≈ 0.0843, Sdp (t) ≈ 0.0293. The crystal field parameters Bkl can be predicted by the empirical superposition model [21]. The parameters Bkl in the superposition model for dn ions in trigonal octahedral system are written as B20 = 6A¯ 2 (R)(3cos2 ˇ − 1) B40 = 6A¯ 4 (R)(35cos4 ˇ − 30cos2 ˇ+3) √ B43 = −12 35A¯ 4 (R)sin3 ˇ cos ˇ

(5)

in which the intrinsic parameter ratio A¯ 2 (R)/A¯ 4 (R) for ions in lots of crystals is found in the range of 8–12 [11,12,22–24] and here we take A¯ 2 (R)/A¯ 4 (R)≈ 10, the mean ratio. ˇ is the angle between the direction of distance R and C3 axis. As in the case of distance R, the angle ˇ in an impurity center may differ from the corresponding angle ˇh in the host crystal owing to the size mismatch substitution in the considered system. We assume ˇ = ˇh + ˇ, where ˇh ≈ 50.96◦ [19] in the host GSGG crystal and ˇ is the angular distortion due to the impurity-caused local lattice relaxation. Thus, in the energy matrix, the three unfixed parameters f , A¯ 4 (R) and ˇ are treated as the adjustable parameters which are determined by fitting the observed crystal field energy levels with the complete diagonalization (of energy matrix) method. The diagonalization calculations for the energy matrix find that the reasonable agreements of crystal field energy levels between calculation and experiment for GSGG: Cr3+ crystal require 3dn

f ≈ 0.7644, A¯ 4 (R) ≈ 1160, ˇ ≈ 3.19◦

(6)

On the basis of the value of f , the Racah parameters B, C, the molecular orbital coefficients N ,  and the spin-orbit parameters ,   obtained from Eqs (3), (4) and (2) are listed in Table 1. The calculated crystal field energy levels of GSGG: Cr3+ crystals in comparison with the available experimental values are given in Table 2.

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Table 1 The Racah parameters B, C (in cm−1 ), molecular obital coefficients N ,  and spin-orbit parameters ,   (in cm−1 ) for (CrO6 )9− octahedral clusters in GSGG: Cr3+ crystal. B

C

Nt

Ne

t

e





787

2943

0.8635

0.8366

0.4280

0.5342

248

219

Table 2 The Crystal field energy levels (in cm−1 ) of GSGG: Cr3+ garnet crystal. Energy levels

Calculation

Oh

D3d

4

A2g

4

A2g

0 0.51

2

Eg (G)

2

Eg

14364 14389

2

T1g (G)

2

Eg

2

A2g

15096 15100 15242

4

Eg

4

A1g

T2g (H)

2

A1g E2g

21201 21606 21697

T1g (F)

4

A2g

4

Eg

22590 22594 22923 22940 22960 22976

A1g (G)

2

A1g

27461

T1g (H)

2

Eg

2

A1g

29697 29729 30090

2

A1g Eg

30156 30418 30529

2

Eg

32175 32176

2

A1g Eg

34930 34954 34975

4

Eg

4

A2g

34985 35599 35634 35708 36387 36426

2

Eg

2

A1g

41362 41377 41481

4

2

4

2 2

T2g (F)

2 a

T

2

T

2

2

Eg (H)

2 b

4

2g

(H)

2g

2

(H)

T1g (P)

T2g (D)

2

15335 15372 15415 15509 15751 15772

2

A2g (F)

2

A1g

43175

2

T1g (P)

2

Eg

2

A2g

45217 45317 45850

2

A1g Eg

2

T2g (F)

2

47087 47564 47666

Experiments

14364[5] 14389[5]

14430[3] 14749[4]

15401[5]

15564[4]

15625[3]

21930[5]

21810[4]

21980[3]

∼34600[6]

∼47600[6]

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3. Discussion The calculations show that there is an angular distortion ˇ (≈3.19◦ ) for Cr3+ centers in GSGG: Cr3+ crystal. In fact, similar angular distortions for Cr3+ -doped other garnet (YA1G, LuA1G and YGG) crystals were found by analysing their spectroscopic data [25]. In view of the size mismatch substitution in Cr3+ -doped GSGG and other garnet crystals, the impurity-caused local lattice relaxation seems to be comprehensible. Table 2 indicates that the seven available observed crystal-field energy levels for GSGG: Cr3+ crystal are rationally explained by the complete diagonalization (of energy matrix) method with only three adjustable parameters. This suggests that the method is efficacious in the investigations of crystal field energy levels and also the defect structure for d3 impurity ions in crystals. The small deviations between the calculated and observed crystal field energy levels may be due to the following reasons: (i) Experimentally, the observed crystal field energy levels differ slightly from sample to sample (see Table 2), specifically, the optical spectral experiments found that there are four distinct Cr3+ centers with slightly different first excited state energy levels in GSGG: Cr3+ crystals, which are due to a small degree of disorder in the distribution of Sc3+ and Ga3+ ions among the sites in GSGG garnet crystals [7]. (ii) Some theoretical treatments, e. g., the superposition model in the estimation of crystal field parameters, are approximate. (iii) The observed crystal field energy levels originate from two contributions, the static one due to the crystal field and the vibrational one from the electron-phonon interaction [26,27], whereas in the theoretical calculations, the small vibrational contributions are omitted. 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