United calculation of the optical and EPR spectral data for Co2+-doped CdS crystal

Optik - International Journal for Light and Electron Optics 194 (2019) 163087

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Original research article

United calculation of the optical and EPR spectral data for Co2+doped CdS crystal

T

Jian Hea,b, Yang Meia,b, , Wen-Chen Zhengc ⁎

a b c

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

ARTICLE INFO

ABSTRACT

Keywords: Optical spectra Electron paramagnetic resonance Crystal- and ligand-field theory Co2+ CdS

This article reports a united calculation of the optical band positions and EPR parameters (g factors g//, g⊥ and zero-field splitting D) for Co2+ ion at the trigonal tetrahedral Cd2+ site of CdS crystal by using the complete diagonalization (of energy matrix) method built on the cluster approach. In the approach, the one-electron basis functions are the molecular orbitals composed of the d orbitals of central dn ion and p orbitals of ligands (rather than only the pure d orbitals in the conventional crystal-field theory), and so the contributions from not only the spin-orbit parameter of dn ion, but also that of ligand ion are contained. The calculated results are in rational agreement with the twenty-two (nineteen optical band positions and three EPR parameters) observed spectral values found in the literature. The defect structure (particularly, the angular distortion) of the trigonal Co2+ center in CdS: Co2+ is also acquired by the calculation. The outcomes are discussed.

1. Introduction II-VI semiconductors (such as ZnS, ZnSe, CdS, CdSe, ZnO) doped with divalent 3dn transition metal ions (e. g., Cr2+, Mn2+, Fe2+, Co , Ni2+) exhibit infrared luminescence and can be applied as laser materials [1–3]. They are also the well-known diluted magnetic semiconductors (DMSs) and hence have the extensive potential applications in spintronics, magnetic recording, magnetic switching and photoelectronic devices [4–8]. These important applications lead their spectroscopic properties to capture considerable attention of researchers [1–5,9–15]. For example, the optical and electron paramagnetic resonance (EPR) spectra of Co2+ doped CdS crystal (with the wurtzite structure) were experimentally studied by several groups [5,11,10–15]. In these studies, nineteen optical band positions and three EPR (or spin-Hamiltonian) parameters (g factors g//, g⊥ and zero-field splitting D) were reported. These spectroscopic data represent unambiguously that Co2+ ion replaces Cd2+ ion to form a trigonal tetrahedral cluster (CoS4)6― in CdS. Theoretically, Jugessur et al. [12] calculated part of optical band positions (i. e., only those related to spin-allowed transitions) and EPR parameters for the trigonal Co2+ center in CdS with the molecular orbital approach, however, the calculated EPR parameters (g// ≈ 2.1427, g⊥≈ 2.1434, D ≈ 0.09 cm−1 [12]) are in poor agreement with the experimental values (g// ≈ 2.269, g⊥≈ 2.286, D ≈ 0.66 cm−1 [15]). Zheng et al. [16]. calculated only the EPR parameters from the high-order perturbation formulas based on the twospin-orbit parameter model. So far, no satisfactory and united theoretical calculations for all these optical and EPR spectral data of CdS: Co2+ have been conducted. Since the complete diagonalization (of energy matrix) method is an effective method to calculate the optical and EPR spectral data in a unified way for dn ions in crystals [17–19], in this article, we compute simultaneously all these 2+

⁎

Corresponding author at: School of Mathematics & Physics, Mianyang Teachers’ College, Mianyang 621000, PR China. E-mail address: [email protected] (Y. Mei).

https://doi.org/10.1016/j.ijleo.2019.163087 Received 26 April 2019; Accepted 10 July 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 194 (2019) 163087

J. He, et al.

optical and EPR spectral data for the trigonal Co2+ center in CdS: Co2+ crystal by the use of the complete diagonalization (of energy matrix) method resting on the cluster approach. In the approach [16,18,19], the one electron basis functions are the molecular orbitals composed of the d orbitals of dn ion and p orbitals of ligand (rather than only the pure d orbitals in the traditional crystal field theory) and so both the contributions of spin-orbit parameter of dn ion and that of ligand ion are contained (so it is also called twospin-orbit-parameter model). The defect structure (which can influence sensitively the EPR data) of the trigonal (CoS4)6- center in CdS is also evaluated. The outcomes are discussed. 2. Calculation Because the one-electron basis functions in the cluster approach are molecular orbitals, we have two spin-orbit parameters ζ, ζ′ and two orbit reduction factors k, k′ [16,19]

= Nt2

0 d

+

0 d

= Nt Ne

2

+

2

ζd0

2

2 )2

)2

(

k = Nt Ne 1 +

)2

(

+ (

k = Nt2 1

)2

(

2

2

2

+

2

0 p

0 p

+2

Sdp ( ) + 2

+4

Sdp ( ) +

Sdp ( )

Sdp ( )

ζp0 are −1

(1) ζd0(Co2+)

n

−1

in which and the spin-orbit parameters of d ion and ligand ions in free state. Here we have ≈ 533 cm [20] and ζd0(S2-) ≈ 365 cm [21] for (CoS4)6- cluster studied. The group overlap integrals Sdp(β) (β = σ or π) are commonly estimated from the Slater-type self-consistent field (SCF) functions [22,23] with the mean metal-ligand distance R¯ . The distinction between the ionic radii ri of impurity ion and rh of the replaced host ion leads the distance R in an impurity center to be unlike the corresponding 1 distance Rh in the host crystal. An empirical equation R = Rh + 2 (ri rh) [24] is applied to estimate the distance R¯ . From ri(Co2+) ≈ 2+ 0.72 Å, rh(Cd ) ≈ 0.94 Å [25], Rh1 ≈ 2.538 Å and Rh2 ≈ 2.525 Å [15] of CdS, we obtain R1 ≈ 2.428 Å, R2 ≈ 2.415 Å and then R¯ ≈ 1 (R1+3R2) ≈ 2.418 Å for (CoS4)6- cluster in CdS: Co2+. Accordingly, we find Sdp(π) ≈ 0.0065 and Sdp(σ) ≈ -0.0276 for the (CoS4)64 cluster in CdS: Co2+ from the Slater-type SCF functions [22,23]. Nγ (γ = t or e) and λβ are the molecular orbital coefficients. They can be computed by the normalization relations [19]

Ne2 [1 + 3 Nt2 [1 +

2

2

+6

+

2

Sdp ( )] = 1 +2

Sdp ( ) + 2

(2)

Sdp ( )] = 1

and the approximate connections [19]

fe = Ne4 [1 + 6

Sdp ( ) + 9

2

2 Sdp ( )]

ft = Nt4 [1 + 2 Sdp ( ) + 2 Sdp ( ) + 2

+

2

2 Sdp ( )+

2

Sdp ( ) Sdp ( )

2 Sdp ( )]

(3) 1 2

(

B B0

C C0

)

+ , B and C (which are often treated as adjustable parameters) are the Racah where the covalence factor ft ≈ fe ≈ fγ ≈ parameters of dn ions in the crystal considered, and B0 and C0 are those of dn ions in free state. For free Co2+ ion, we find B0 ≈ 1115 cm−1 and C0 ≈ 4366 cm−1 [20]. The Hamiltonian of a trigonal d7 tetrahedral clusters in crystals in the cluster approach contains the Coulomb, spin-orbit and crystal field interaction terms, i. e., H = HCoul.(B, C) + HSO(ζ, ζ′) + HCF(B20, B40, B43)

(4)

The full energy matrix of the Hamiltonian is 120 × 120 dimensions and can be built by the strong field basis functions [26]. The eigenvalues of the energy matrix are equivalent to the crystal field energy levels and hence to the optical band positions. The EPR parameter are computed from the eigenvectors |4A2, Ms〉 and eigenvalues E(4A2, Ms) of the ground state 4A2 in terms of the equations: 4

g// = 2 g =

4

A2 ,

A2 ,

1 2

1 2

4

(k , k ) LZ + gS SZ (k, k ) LX + gS SX

4

A2 ,

A2 ,

1 2 1 2 2

Optik - International Journal for Light and Electron Optics 194 (2019) 163087

J. He, et al.

Table 1 The crystal field energy levels (or optical band positions, in cm−1) of the trigonal (CoS4)6- tetrahedral clusters in CdS: Co2+ crystal. Symmetry Oh 4 A2

Calculation Experiment Td A2

4

4

T2g

4

T1g (F)

4

Eg

4

A2g

4

Eg

4

A2g

2

Eg

2

A1g Eg

0 1.329 3138 3189 3218 3297 3304 3381 5377 5476 5527 5861 5956 6142 12264 12270 12767 12810 12848 13680 13786 13820 14175 14207 14224 14351 14465 14861 14910 16485 16600 16606 16899 17061 17063 18212 18213 18880 19000 19066

2

Eg(G)

2

T1g (P)

4

T1g(P)

4

4

A1g

2

Eg

2

A2g Eg

2

A1g(G) T2g(G)

2

2

T2g(G)

2 2

2

T1ga(H)

2 2

2

2

2

2

Eg(H) T1gb(H)

a

2

A2g Eg A1g Eg A1g Eg

0 1.32 [15]a 3300 [11,12]

5470 5520 5530 5604 6135 6496

[11,12] [11,12] [11,12] [11,12] [11,12] [11,12]

5500 [13]

13045 [11,12] 13154 13321 13541 13618 13847 14142

[11,12] [11,12] [11,12] [11,12] [11,12] [11,12]

13,717 [14] 13,495 [14] 13,950 [13]

14,598 [14]

14,462 [5] 14749 [5] 15071 [5] 15256 [5]

16650 [13]

19120 [14]

Obtained from EPR spectra.

D=

1 3 E 4 A2 ± 2 2

E 4 A2 ±

1 2

(5)

Thereby, the optical and EPR spectral data can be calculated in a unified way by diagonalizating the complete energy matrix. In the energy matrix, the crystal field parameters Bkl are usually evaluated by means of the superposition model [27] in which the parameters Bkl for a trigonal dn tetrahedral cluster are given as

R R B20 = A¯2 (R 0)[2( 0 )t2 + 3( 0 )t2 (3 cos2 R1 R2 R R B40 = A¯ 4 (R0)[8( 0 )t4 + 3( 0 )t4 (35 cos4 R1 R2

B43 = 6 35 A¯ 4 (R 0)(

1)] 30 cos2

+ 3)]

R 0 t4 3 ) sin cos R2

(6)

in which the intrinsic parameter ratio A¯ 2 (R 0) / A¯ 4 (R 0) is found to be the range of 8˜12 [16–19,27–29] for 3dn ions in lots of crystals, and the mean ratio A¯ 2 (R 0) / A¯ 4 (R 0) ≈ 10 is used here. R0 (=R¯ ) is the reference distance. The power-law exponents t2 ≈ 3 and t4 ≈ 5 [16,19,27–29]. θ is the included angle of the bond lengths R1 (only C3 axis) and R2. The bond angle θ in an impurity center and the corresponding host angle θh may be not alike, as in the case of bond lengths. Thus, we take θ ≈ θh+Δθ, where θh ≈ 108.93˚ [15] in the host CdS and Δθ represents the impurity-caused angular distortion. Consequently, in the energy matrix, there are four unfixed or adjustable parameters B, C, A¯ 4 (R 0) and Δθ to be determined by fitting the observed spectroscopic data. The agreements between the computed optical and EPR spectral data (from the above complete diagonalization method) and the experimental values 3

Optik - International Journal for Light and Electron Optics 194 (2019) 163087

J. He, et al.

Table 2 The EPR (or spin-Hamiltonian) parameters for the trigonal Co2+ center in CdS crystal. g//

D (cm−1)

g⊥

Calc. 2.215

Expt. [15] 2.269

Calc. 2.228

Expt. [15] 2.286

Calc. 0.67

Expt. [15] 0.66 (2)

require these adjustable parameters to be B ≈ 658 cm−1, C ≈ 2750 cm−1, A¯ 4 (R 0) ≈ 530 cm−1, Δθ ≈ 1.06°

(7)

The computed optical and EPR spectral data compared with the observed values are given in Table 1 and 2, respectively. 3. Discussion The impurity-caused angular distance Δθ ≠ 0 obtained from the above calculation indicates that like the case of bond lengths Ri, the bond angle θ in the Co2+ impurity center is also dissimilar to the host angle θh owing to the size mismatch substitution in CdS: Co2+. This angular distortion Δθ leads the angle θ (≈ 109.99˚) in the Co2+ center to be closer to θ0 (≈109.47˚, the same angle in cubic tetrahedron) than θh (≈ 108.93˚ [15]) in CdS, suggesting that the trigonal distortion of Co2+ center in CdS: Co2+ is small. This explains the small observed zero-field splitting D in CdS: Co2+ [15]. So, the angular distortion is reasonable. It seems that information on the defect structure of a dn impurity center in crystals can be attained by analyzing its spectroscopic data. Tables 1 and 2 state clearly that by employing only four adjustable parameters, the calculated optical and EPR spectral data of CdS: Co2+ crystal from the complete diagonalization (of energy matrix) method based on the cluster approach are in rational agreement with the twenty-two available observed values (nineteen optical bands and three EPR parameters). The small differences between the calculated and experimental values may be due mainly to the cause: The observed spectroscopic data stem from two factors, the static factor owing to static crystal field and the dynamic factor originated from electron-phonon interaction [30–32], but in the above calculation, we disregard the small dynamic factor. So, the above computed outcomes can be seen as suitable, and the complete diagonalization (of energy matrix) method has the capacity to analyze unitedly the optical and EPR spectral data for a dn ion in crystals. 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