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Theoretical study of spin singlets contributions to zero-field splitting and local lattice structure of Cr2+ in CdGa2S4
Journal of Alloys and Compounds 484 (2009) 472–476
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Theoretical study of spin singlets contributions to zero-field splitting and local lattice structure of Cr2+ in CdGa2 S4 Yang Li a , Xiao-Yu Kuang a,b,∗ , Zhe Li a , Ying Li a , Ming-Liang Gao a a b
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China International Center of Materials Physics, Academia Sinica, Shengyang 110016, China
a r t i c l e
i n f o
Article history: Received 11 June 2008 Accepted 28 April 2009 Available online 5 May 2009 Keywords: Crystal structure and symmetry Crystal and ligand fields EPR
a b s t r a c t A theoretical method for investigating the inter-relation between electronic and the molecular structures of a d4 ion in a tetragonal ligand-field has been established on the basis of 210 × 210 complete energy matrix within a weak-field-representation. Using the method, the local structure parameters of CdGa2 S4 :Cr2+ system are determined by the experimental EPR zero-field splitting (ZFS) spectra. Our results show that the local structure around Cr2+ is a compression distortion and the local lattice structure parameters R = 2.46 Å and = 57.63◦ are determined. Moreover, the contributions of the spin singlets to ZFS parameters of Cr2+ ions in CdGa2 S4 crystals are investigated for the first time. The results indicate that the spin singlets contributions to ZFS parameter D are negligible, but the contributions to ZFS parameters a and F cannot be neglected. © 2009 Published by Elsevier B.V.
1. Introduction Impurities in semiconductors have attracted a great deal of attention for many years owing to their significance for practical applications, as in photoconductors, microwave detectors, and electroluminescent devices [1–8]. Among the impurities, particular attention has been focused on the transition metal ions because they are commonly associated with deep levels within the host crystal band gap. Cadmium thiogallate, CdGa2 S4 , belongs to a wide class of AII B2 III C4 VI ternary semiconductors, which has been extensively studied in recent years [9–12]. It is known that the introduction of transition metal ions, particularly Cr2+ , into II–VI materials such as ZnSe and ZnS, has made it possible to extend their capabilities and to produce broadly tunable lasers [13]. Recently investigations demonstrate that CdGa2 S4 doped with Cr2+ may also be a laser-active material [12]. The EPR spectra of transition metal Cr2+ ions doped into CdGa2 S4 have been experimentally observed by Avanesov et al. [11]. Their experimental results give important information about the ground state of the transition metal Cr2+ ions and form a useful starting point for understanding the interrelationships between electronic and molecular structure of Cr2+ ions in (CrS4 )6− coordination complex. Despite the large number of publications relating to Cr2+ ions in a CdGa2 S4 :Cr2+ system, as yet a
∗ Corresponding author at: Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China. Tel.: +86 028 85403803; fax: +86 028 85405515. E-mail address: scu [email protected] (X.-Y. Kuang). 0925-8388/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.jallcom.2009.04.135
comprehensive report of its zero-field splitting parameters is lacking. Theoretically, the studies of the electronic structure of transition metal Cr2+ impurities in crystals have made remarkable progress in the past decades by the 5 D approximation [14–16]. However, we have not fully been able to understand the nature of transition metal Cr2+ ions within this approach because the contributions of spin triplet states 3 L (L = H, G, F, D, P) and the spin singlets states 1 L (L = I, G, F, D, S) have been neglected in them. To remedy these discrepancies between theory and experiments, the spin triplet states contributions to the zero-field splitting (ZFS) for a d4 configuration ion in crystals were performed by Zhou et al. [17–19]. Unfortunately, these method are still insufficient to understand the detailed information and physical origin of transition metal Cr2+ ions in crystals because the spin singlets states 1 L (L = I, G, F, D, S) influence the fine structure splitting of the ground states, i.e., affect the ground zero-field splitting parameters. It is well known that the Hamiltonian matrix of a d4 configuration in crystals has 210 × 210 dimensions for all the spin states but only 25 × 25 for the 5 D state and 160 × 160 for both 5 D and 3 L states. So, to obtain more accurate ZFS, all 2S+1 L multiplets with S = 2, 1 and 0 should be considered, i.e., a complete calculation. In this paper, complete energy matrices (210 × 210) of a d4 configuration ion in a tetragonal ligand-field are constructed and the ZFS parameters a, D and F of the CdGa2 S4 :Cr2+ system are investigated. By diagonalizing the complete energy matrices, the local structure distortion parameters R and are determined. Moreover, the contributions of the spin singlets to ZFS parameters of Cr2+ ions in CdGa2 S4 crystals are investigated for the first time.
Y. Li et al. / Journal of Alloys and Compounds 484 (2009) 472–476
473
Table 1 Spin Hamiltonian matrix. S,MS
2,2
2,2
2D +
2,−2
a 2
2,−2 a 10
+
F 15
a 2
2D +
a 10
+
F 15
2,0
2,1
2,−1
0
0
0
0
0
0
3a 5
+
2F 5
2,0
0
0
−2D +
2,1
0
0
0
−D −
2,−1
0
0
0
0
For the CdGa2 S4 :Cr2+ system, the local symmetry of the centers is tetragonal. The EPR spectrum of the tetragonal Cr2+ may be described in terms of the following spin Hamiltonian [20]:
+
a (35SZ4 − 155SZ2 + 72) 120
F a 4 4 )+ (S + S− (35SZ4 − 155SZ2 + 72) 48 + 180
(1)
where a, D and F are the so-called zero-field splitting parameters. a is the cubic field splitting parameter, D and F correspond to axial component of the second-order and thefourth-order, respectively. By combing the effective spin function SM for S = 2, we can construct a set of spin basis functions of the irreducible representations (A1 , A2 , B2 , E) of the 5 B2 ground state for spin Hamiltonian as follows:
A1 = √i (2 − 2 − 22 ) 2
A2 = √1 (2 − 2 + 22 ) 2
Ey = √i (21 + 2 − 1 )
(2)
2 1 Ex = √ (21 − 2 − 1 ) 2
B2 = 20
F 2a E(A1 ) = 2D − + 5 15 3a F E(A2 ) = 2D + + 5 15 4F 2a − ( = x, y) E(E ) = −D − 5 15 3a 2F E(B2 ) = −2D + + 5 5
4F 15
0 −D −
2a 5
−
4F 15
ˆ =H ˆ ee + H ˆ SO + H ˆ LF = H
e2 i
rij
+
li · si +
i
Vi
(6)
i
ˆ SO denotes the ˆ ee denotes the electrostatic energy, H where H ˆ LF denotes the ligand-field energy. spin–orbit coupling energy and H ς is the spin–orbit coupling coefficient, and Vi is the ligand-field potential: Vi = 00 Z00 + 20 ri2 Z20 (i , i ) + 40 ri4 Z40 (i , i ) c 4 c s 4 S ri Z44 (i , i ) + 44 ri Z44 (i , i ) + 44
(7)
where ri , i and i are spherical coordinates of the ith electron. Zlm , c and Z s are defined as: Zlm lm Zl0 = Yl0 m c = 1 [Y Zlm √ l,−m + (−1) Yl,m ] 2 m S = i [Y Zlm √ l,−m − (−1) Yl,m ] 2
(8)
c and s The Ylm in Eq. (8) are the spherical harmonics. l0 , lm lm 4 are associated with the local lattice structure of the 3d ion by the relations:
4 eq
Zl0 ( , ) 2l + 1 R l+1
b44 =
5a 2
4 eq c Zlm ( , ) 2l + 1 R l+1 n
c =− lm
(9)
4 eq s Zlm ( , ) 2l + 1 R l+1 n
=1
(3)
where (R , , ) are the spherical coordinates of the th ligand, q is its effective charge. The matrix elements of Hamiltonian (6) are functions of the Racah parameters B and C, the spin–orbit coupling coefficient , and the ligand-field parameters which are generally expressed as follows [21]: (4)
B20 = B40 =
The values of E( ) can be obtained by comparison with the eigenvalues of the d4 matrix corresponding to the orbitally nondegenerate ground state. It is noteworthy to mention that the q parameters a, D and F are related to the ZFS parameters bk . The relationships are given by a F + , 2 3
=1
s =− lm
a = E(A2 ) − E(A1 ) 1 D = − (E(E ) − E(A1 ) − E(A2 ) + E(B2 )) 7 3 F = (3E(B2 ) − 3E(A2 ) − 4E(E ) + 4E(A1 )) 7
l0 = −
=1
Thus, we have
b04 =
−
n
With such a set of spin basis functions, the ZFS Hamiltonian matrix reduces into diagonal blocks, shown in Table 1. From this, it is very easy to obtain the eigenvalues of Eq. (1),
b02 = D,
0 2a 5
The Hamiltonian for a 3d4 configuration ion in a tetragonal ligand-field can be written as:
2. Theoretical model
ˆ S = D(S 2 − 2) + H Z
0
(5)
c B44
=
5 1/2 4
9 1/2 4
9 1/2
s =i B44
8
20 r 2
40 r 4 c 44
9 1/2 8
4
(10)
r
s 44 r4
For the CdGa2 S4 :Cr2+ system, the local structure symmetry belongs to the group D2d . Taking the superposition model, the
474
Y. Li et al. / Journal of Alloys and Compounds 484 (2009) 472–476 Table 2 Zero-field splitting parameters a, D and F for Cr2+ ions in CdGa2 S4 :Cr2+ system as a function of the three parameters R, , and N. N
R (Å)
(◦ )
a
D
F
0.985 0.985 0.985 0.965 0.965 0.965 0.945 0.945 0.945 0.925 0.925 0.925 Expt [11]
−0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03
2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70
0.042 0.049 0.061 0.042 0.048 0.059 0.042 0.048 0.58 0.043 0.048 0.057 0.043
−2.48 −2.54 −2.56 −2.39 −2.44 −2.46 −2.32 −2.36 −2.37 −2.25 −2.28 −2.29 −2.37
−0.042 −0.047 −0.056 −0.045 −0.049 −0.057 −0.047 −0.051 −0.060 −0.050 −0.054 −0.061 −0.054
3. Calculations for Cr2+ ions in the CdGa2 S4 :Cr2+ system 3.1. The local lattice structure distortion of Cr2+ ions in CdGa2 S4 Fig. 1. The local structure distortion of CdGa2 S4 :Cr system. R and denote the Cr–S bond length and angle between Cr–S bond and Z-axis. R0 and 0 are the Cd–S bond length and the angle between Cd–S bond and Z-axis of the host crystal CdGa2 S4 R and denote the structure distortion in CdGa2 S4 :Cr2+ system. 2+
s will vanish (the coordinates are chosen liangd-field parameters B44 as shown in Fig. 1) and the rest can be written as:
A2 (3 cos2 − 1) R3 1 A4 = (35 cos4 − 30 cos2 + 3) 2 R5 √ A4 = 70 5 sin4 R
B20 = 2 B40 c B44
where
A4 = −eq r 4 ,
(11)
The experimental ZFS data for the CdGa2 S4 :Cr2+ system have been reported [11]. From the experimental spectra, we can study the local geometric structure by diagonalizing the complete energy matrix. When Cr2+ doped in CdGa2 S4 host crystal, a Cr2+ will substitute for a Cd2+ ion, the surrounding local structure displays a tetragonal distortion, which can be described by two parameters R and (see Fig. 1). If one uses R0 and 0 to represent the Cd–S bond length and the angle between Cd–S bond and Z-axis of the host crystal CdGa2 S4 , respectively, then the local structure parameters R and for CdGa2 S4 :Cr2+ system may be expressed as R = R0 + R,
A2 = −eq r 2
(12)
R and denote the Cr–S bond length and angle between Cr–S bond and Z-axis, respectively, q the charge of ligand, and −e is the electron charge. The ratio r2 /r4 = 0.12205 a.u. can be calculated from the radial wave function [22]. A4 is almost a constant for the (CrS4 )6− tetrahedron, and can be determined from the absorption spectrum and the Cr–S bond length of the CrS crystal. For CrS, from its spectrum [23], we derive A4 = 57.6781 a.u. and A2 = 7.0395 a.u. for the (CrS4 )6− tetrahedron, and we will use them in the following calculations. Using the complete energy matrices and Eq. (11), the inter-relation between the local lattice structure parameters R and of CdGa2 S4 :Cr2+ system and ZFS parameters a, D and F can be established.
= 0 + ,
= 0 +
(13)
where R0 = 2.52 Å and 0 = 55.64◦ [24]. To decrease the number of adjustable parameters and reflect the covalency effects, we use the Curie et al.’s covalent theory and take an average covalence factor N to determine the optical parameters as following [25]: B = N 4 B0 ,
B = N 4 B0 ,
= N 2 0
(14)
where the free Cr2+ ion parameters [22] are B0 = 870.6 cm−1 , C0 = 3135.7 cm−1 and 0 = 232 cm−1 . For Cr2+ ions in CdGa2 S4 :Cr2+ system, by diagonalizing the complete energy matrices, the groundstate ZFS can be calculated with use of distortion parameters R and and covalence factor N. The comparisons between the theoretical values and experimental findings are shown in Table 2. It can be seen from Table 2 that the distortion parameters R = −0.06 Å and = 1.99◦ and the covalence factor N = 0.945 can provide a satisfactory explanation for the experimental ZFS parameters a, D and F. R < 0 indicates that the local lattice structure of
Table 3 Zero-field splitting parameters of CdGa2 S4 :Cr2+ system as a function of R, , and N. N
R (Å)
(◦ )
(a)a
(D)a
(F)a
(a)b
(D)b
(F)b
0.985 0.985 0.985 0.965 0.965 0.965 0.945 0.945 0.945 0.925 0.925 0.925
−0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03
2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70
0.036 0.043 0.055 0.035 0.041 0.052 0.034 0.040 0.049 0.033 0.038 0.047
−2.48 −2.54 −2.56 −2.39 −2.44 −2.46 −2.31 −2.35 −2.37 −2.25 −2.28 −2.29
−0.031 −0.036 −0.045 −0.032 −0.028 −0.022 −0.032 −0.037 −0.043 −0.033 −0.036 −0.043
0.007 0.010 0.015 0.006 0.008 0.013 0.005 0.007 0.011 0.004 0.006 0.009
−1.31 −1.37 −1.39 −1.21 −1.26 −1.28 −1.11 −1.16 −1.17 −1.02 −1.06 −1.07
−0.001 −0.002 −0.001 −0.001 −0.001 −0.005 −0.001 −0.002 −0.004 0 −0.001 −0.003
a b
Neglecting the spin singlets. Neglecting the spin singlets and the spin triplets.
Y. Li et al. / Journal of Alloys and Compounds 484 (2009) 472–476
475
Fig. 2. Spin singlets and spin triplets contributions to the fourth-order zero-field splitting parameters a and F of the CdGa2 S4 :Cr2+ system. Curve 1: neglecting the spin singlets; Curve 2: neglecting both the spin singlets and the spin triplets. Table 4 Spin singlets and spin triplets contributions to the zero-field splitting parameters a, D and F of CdGa2 S4 :Cr2+ system. N
R (Å)
(◦ )
(ra )a
(rD )a
(rF )a
(ra )b
(rD )b
(rF )b
0.985 0.985 0.985 0.965 0.965 0.965 0.945 0.945 0.945 0.925 0.925 0.925
−0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03
2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70
0.1429 0.1224 0.0984 0.1667 0.1458 0.1186 0.1905 0.1667 0.1724 0.2326 0.2083 0.1715
0 0 0 0 0 0 0.0043 0.0042 0.0042 0 0 0
0.2619 0.2340 0.1964 0.2889 0.4288 0.6146 0.3191 0.2745 0.2833 0.3400 0.3333 0.2951
0.8333 0.7959 0.7541 0.8571 0.8333 0.7797 0.8810 0.8542 0.8103 0.9070 0.8750 0.8421
0.4718 0.4606 0.4570 0.4937 0.4836 0.4797 0.5216 0.5085 0.5063 0.5467 0.5351 0.5328
0.9762 0.9574 0.9821 0.9778 0.9796 0.9123 0.9787 0.9608 0.9333 1 0.9815 0.9508
a b
Neglecting the spin singlets. Neglecting the spin singlets and the spin triplets.
CdGa2 S4 :Cr2+ system has a compression distortion. The compression distortion may be ascribed to the fact that the radius of Cr2+ ion (r = 0.89 Å) is smaller than that of Cd2+ ion (r = 0.97 Å) [26]. From the calculation, the local lattice structure parameters R = 2.46 Å and = 57.63◦ for Cr2+ ions in CdGa2 S4 :Cr2+ system have been determined. Of course, careful experimental investigations, especially ENDOR experiments, are required in order to clarify the local lattice structure around the Cr2+ ions in CdGa2 S4 :Cr2+ system in detail. 3.2. The spin singlets contributions to ZFS of Cr2+ ions in CdGa2 S4 For further studying of the spin singlets contribution to the ZFS parameters of CdGa2 S4 :Cr2+ system, the ratios
a − a , a
ra =
D − D , D
rD =
F − F F
rF =
(15)
versus R, and N are calculated (Table 4), where a, D and F are the ZFS parameters considering all the spin states, and a , D and F those neglecting the spin singlets or neglecting both the spin singlets and spin triplets. ZFS parameters a , D and F are calculated can be seen in Table 3. The detailed results are presented in Figs. 2 and 3. It is obviously that the larger the ratio r, the larger are the contributions to the ZFS parameters. From Figs. 2 and 3, we can see that the spin singlets contributions to the ZFS parameter D are negligible, but the contributions to the ZFS parameters a and F cannot be neglected. It can be also seen from Figs. 2 and 3 that the values r for 5 D approximation is more than 0.46 for ZFS parameter D and 0.66 for ZFS parameter a and F. This means that the 5 D approximation method is imperfect. A part of the quantitative calculation results are listed in Tables 3 and 4.
Fig. 3. Spin singlets and spin triplets contributions to the second-order zero-field splitting parameter D of the CdGa2 S4 :Cr2 system. (Curve 1) Neglecting the spin singlets; (Curve 2) neglecting both the spin singlets and the spin triplets.
4. Conclusions A theoretical method for studying the inter-relation between molecular structure and electronic structure has been proposed by diagonalizing the complete energy matrices. From the above studies, we have the following conclusions: (i) The zero-field splitting parameters a, D and F of Cr2+ in CdGa2 S4 :Cr2+ system have been studied by the 210 × 210 complete energy matrix for a d4 configuration ion in tetragonal
476
Y. Li et al. / Journal of Alloys and Compounds 484 (2009) 472–476
ligand-field. By diagonalizing the complete energy matrix, the local structure distortion parameters R = −0.06 Å and = 1.99◦ are determined. The results show that the local lattice structure of (CrS4 )6− coordination complex for Cr2+ ions in a CdGa2 S4 :Cr2+ system has an compression distortion. This distortion may be ascribed to the fact that the radius of Cr2+ ion (r = 0.89 Å) is smaller than that of Cd2+ ion (r = 0.97 Å). (ii) The spin singlets contributions to the zero-field splitting parameters of Cr2+ ions in CdGa2 S4 :Cr2+ system are investigated for the first time. It is shown that the spin singlets contributions to the zero-field splitting parameters a and F are important, and it cannot be neglected in general calculation. Acknowledgements This project was supported by National Natural Science Foundation of China (No. 10774103) and the Doctoral Education fund of Education Ministry of China (No. 20050610011). References [1] D. Haranath, H. Chander, N. Bhalla, P. Sharma, K.N. Sood, Appl. Phys. Lett. 86 (2005) 201904. [2] F.H. Su, Z.L. Fang, B.S. Ma, K. Ding, G.H. Li, W. Chen, J. Phys. Chem. B 107 (2003) 6991.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
J.H. Chung, C.S. Ah, D.J. Jang, J. Phys. Chem. B 105 (2001) 4128. A.A. Bol, A. Meijerink, J. Phys. Chem. B 105 (2001) 10197. A.A. Bol, A. Meijerink, J. Phys. Chem. B 105 (2001) 10203. Z. Quan, Z. Wang, P. Yang, J. Lin, J. Fang, Inorg. Chem. 46 (2007) 1354. P.A. Gonzalez Beermann, B.R. McGarvey, S. Muralidharan, R.C.W. Sung, Chem. Mater. 16 (2004) 915. Y.R. Do, Y.C. Kim, S.H. Cho, D.S. Zang, Y.D. Huh, S.J. Yun, Appl. Phys. Lett. 84 (2004) 1377. C.Z. Hadad, S.O. Vasquez, Phys. Rev. B 60 (1999) 8586. X.S. Jiang, W.R.L. Lambrecht, Phys. Rev. B 69 (2004) 035201. A.G. Avanesov, V.V. Badikov, G.S. Shakurov, Phys. Solid State 45 (2003) 1451. M.B. Johnson, S.B. Mirov, V. Fedorov, M.E. Zvanut, J.G. Harrison, V.V. Badikov, G.S. Shevirdyaeva, Opt. Commun. 233 (2004) 403. L. Deloach, R. Page, G. Wilke, S. Payne, W. Krupke, IEEE J. Quant. Electron. 32 (1996) 885. C. Rudowicz, Y.Y. Zhou, L.W. Yu, J. Phys. Chem. Solids 53 (1992) 1227. C. Rudowicz, Y.Y. Zhou, J. Magn. Magn. Mater. 111 (1992) 153. C. Rudowicz, M.L. Du, Y.Y. Yeung, Y.Y. Zhou, Physica B 191 (1993) 323. Y.Y. Zhou, C.L. Li, Phys. Rev. B 48 (1993) 16489. Y.Y. Zhou, Phys. Rev. B 50 (1994) 7161. Z.F. Li, D.H. Li, Y.Y. Zhou, Physica B 252 (1998) 167. U. Kaufmann, Phys. Rev. B 14 (1976) 1848. D.J. Newman, W. Urban, Adv. Phys. 24 (1975) 793. M.G. Zhao, L.H. Xie, Mater. Sci. Eng. B 75 (2000) 72. T.C. De Vore, Ph.D. Thesis, Iowa State University, USA 1975. A.A. Lavrentyev, B.V. Gabrelian, V.A. Dubeiko, I.Ya. Nikiforov, J.J. Rehr, J. Phys. Chem. Solids 63 (2002) 227. D. Curie, C. Barthon, B. Canny, J. Chem. Phys. 61 (1974) 3048. Z.M. Zhou, Y. Xu, Z.M. Wang, H.D. Jia, Concise Inorganic Chemistry, Zhengzhou University Press, China, 2002.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Theoretical study of spin singlets contributions to zero-field splitting and local lattice structure of Cr2+ in CdGa2 S4 Yang Li a , Xiao-Yu Kuang a,b,∗ , Zhe Li a , Ying Li a , Ming-Liang Gao a a b
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China International Center of Materials Physics, Academia Sinica, Shengyang 110016, China
a r t i c l e
i n f o
Article history: Received 11 June 2008 Accepted 28 April 2009 Available online 5 May 2009 Keywords: Crystal structure and symmetry Crystal and ligand fields EPR
a b s t r a c t A theoretical method for investigating the inter-relation between electronic and the molecular structures of a d4 ion in a tetragonal ligand-field has been established on the basis of 210 × 210 complete energy matrix within a weak-field-representation. Using the method, the local structure parameters of CdGa2 S4 :Cr2+ system are determined by the experimental EPR zero-field splitting (ZFS) spectra. Our results show that the local structure around Cr2+ is a compression distortion and the local lattice structure parameters R = 2.46 Å and = 57.63◦ are determined. Moreover, the contributions of the spin singlets to ZFS parameters of Cr2+ ions in CdGa2 S4 crystals are investigated for the first time. The results indicate that the spin singlets contributions to ZFS parameter D are negligible, but the contributions to ZFS parameters a and F cannot be neglected. © 2009 Published by Elsevier B.V.
1. Introduction Impurities in semiconductors have attracted a great deal of attention for many years owing to their significance for practical applications, as in photoconductors, microwave detectors, and electroluminescent devices [1–8]. Among the impurities, particular attention has been focused on the transition metal ions because they are commonly associated with deep levels within the host crystal band gap. Cadmium thiogallate, CdGa2 S4 , belongs to a wide class of AII B2 III C4 VI ternary semiconductors, which has been extensively studied in recent years [9–12]. It is known that the introduction of transition metal ions, particularly Cr2+ , into II–VI materials such as ZnSe and ZnS, has made it possible to extend their capabilities and to produce broadly tunable lasers [13]. Recently investigations demonstrate that CdGa2 S4 doped with Cr2+ may also be a laser-active material [12]. The EPR spectra of transition metal Cr2+ ions doped into CdGa2 S4 have been experimentally observed by Avanesov et al. [11]. Their experimental results give important information about the ground state of the transition metal Cr2+ ions and form a useful starting point for understanding the interrelationships between electronic and molecular structure of Cr2+ ions in (CrS4 )6− coordination complex. Despite the large number of publications relating to Cr2+ ions in a CdGa2 S4 :Cr2+ system, as yet a
∗ Corresponding author at: Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China. Tel.: +86 028 85403803; fax: +86 028 85405515. E-mail address: scu [email protected] (X.-Y. Kuang). 0925-8388/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.jallcom.2009.04.135
comprehensive report of its zero-field splitting parameters is lacking. Theoretically, the studies of the electronic structure of transition metal Cr2+ impurities in crystals have made remarkable progress in the past decades by the 5 D approximation [14–16]. However, we have not fully been able to understand the nature of transition metal Cr2+ ions within this approach because the contributions of spin triplet states 3 L (L = H, G, F, D, P) and the spin singlets states 1 L (L = I, G, F, D, S) have been neglected in them. To remedy these discrepancies between theory and experiments, the spin triplet states contributions to the zero-field splitting (ZFS) for a d4 configuration ion in crystals were performed by Zhou et al. [17–19]. Unfortunately, these method are still insufficient to understand the detailed information and physical origin of transition metal Cr2+ ions in crystals because the spin singlets states 1 L (L = I, G, F, D, S) influence the fine structure splitting of the ground states, i.e., affect the ground zero-field splitting parameters. It is well known that the Hamiltonian matrix of a d4 configuration in crystals has 210 × 210 dimensions for all the spin states but only 25 × 25 for the 5 D state and 160 × 160 for both 5 D and 3 L states. So, to obtain more accurate ZFS, all 2S+1 L multiplets with S = 2, 1 and 0 should be considered, i.e., a complete calculation. In this paper, complete energy matrices (210 × 210) of a d4 configuration ion in a tetragonal ligand-field are constructed and the ZFS parameters a, D and F of the CdGa2 S4 :Cr2+ system are investigated. By diagonalizing the complete energy matrices, the local structure distortion parameters R and are determined. Moreover, the contributions of the spin singlets to ZFS parameters of Cr2+ ions in CdGa2 S4 crystals are investigated for the first time.
Y. Li et al. / Journal of Alloys and Compounds 484 (2009) 472–476
473
Table 1 Spin Hamiltonian matrix. S,MS
2,2
2,2
2D +
2,−2
a 2
2,−2 a 10
+
F 15
a 2
2D +
a 10
+
F 15
2,0
2,1
2,−1
0
0
0
0
0
0
3a 5
+
2F 5
2,0
0
0
−2D +
2,1
0
0
0
−D −
2,−1
0
0
0
0
For the CdGa2 S4 :Cr2+ system, the local symmetry of the centers is tetragonal. The EPR spectrum of the tetragonal Cr2+ may be described in terms of the following spin Hamiltonian [20]:
+
a (35SZ4 − 155SZ2 + 72) 120
F a 4 4 )+ (S + S− (35SZ4 − 155SZ2 + 72) 48 + 180
(1)
where a, D and F are the so-called zero-field splitting parameters. a is the cubic field splitting parameter, D and F correspond to axial component of the second-order and thefourth-order, respectively. By combing the effective spin function SM for S = 2, we can construct a set of spin basis functions of the irreducible representations (A1 , A2 , B2 , E) of the 5 B2 ground state for spin Hamiltonian as follows:
A1 = √i (2 − 2 − 22 ) 2
A2 = √1 (2 − 2 + 22 ) 2
Ey = √i (21 + 2 − 1 )
(2)
2 1 Ex = √ (21 − 2 − 1 ) 2
B2 = 20
F 2a E(A1 ) = 2D − + 5 15 3a F E(A2 ) = 2D + + 5 15 4F 2a − ( = x, y) E(E ) = −D − 5 15 3a 2F E(B2 ) = −2D + + 5 5
4F 15
0 −D −
2a 5
−
4F 15
ˆ =H ˆ ee + H ˆ SO + H ˆ LF = H
e2 i
rij
+
li · si +
i
Vi
(6)
i
ˆ SO denotes the ˆ ee denotes the electrostatic energy, H where H ˆ LF denotes the ligand-field energy. spin–orbit coupling energy and H ς is the spin–orbit coupling coefficient, and Vi is the ligand-field potential: Vi = 00 Z00 + 20 ri2 Z20 (i , i ) + 40 ri4 Z40 (i , i ) c 4 c s 4 S ri Z44 (i , i ) + 44 ri Z44 (i , i ) + 44
(7)
where ri , i and i are spherical coordinates of the ith electron. Zlm , c and Z s are defined as: Zlm lm Zl0 = Yl0 m c = 1 [Y Zlm √ l,−m + (−1) Yl,m ] 2 m S = i [Y Zlm √ l,−m − (−1) Yl,m ] 2
(8)
c and s The Ylm in Eq. (8) are the spherical harmonics. l0 , lm lm 4 are associated with the local lattice structure of the 3d ion by the relations:
4 eq
Zl0 ( , ) 2l + 1 R l+1
b44 =
5a 2
4 eq c Zlm ( , ) 2l + 1 R l+1 n
c =− lm
(9)
4 eq s Zlm ( , ) 2l + 1 R l+1 n
=1
(3)
where (R , , ) are the spherical coordinates of the th ligand, q is its effective charge. The matrix elements of Hamiltonian (6) are functions of the Racah parameters B and C, the spin–orbit coupling coefficient , and the ligand-field parameters which are generally expressed as follows [21]: (4)
B20 = B40 =
The values of E( ) can be obtained by comparison with the eigenvalues of the d4 matrix corresponding to the orbitally nondegenerate ground state. It is noteworthy to mention that the q parameters a, D and F are related to the ZFS parameters bk . The relationships are given by a F + , 2 3
=1
s =− lm
a = E(A2 ) − E(A1 ) 1 D = − (E(E ) − E(A1 ) − E(A2 ) + E(B2 )) 7 3 F = (3E(B2 ) − 3E(A2 ) − 4E(E ) + 4E(A1 )) 7
l0 = −
=1
Thus, we have
b04 =
−
n
With such a set of spin basis functions, the ZFS Hamiltonian matrix reduces into diagonal blocks, shown in Table 1. From this, it is very easy to obtain the eigenvalues of Eq. (1),
b02 = D,
0 2a 5
The Hamiltonian for a 3d4 configuration ion in a tetragonal ligand-field can be written as:
2. Theoretical model
ˆ S = D(S 2 − 2) + H Z
0
(5)
c B44
=
5 1/2 4
9 1/2 4
9 1/2
s =i B44
8
20 r 2
40 r 4 c 44
9 1/2 8
4
(10)
r
s 44 r4
For the CdGa2 S4 :Cr2+ system, the local structure symmetry belongs to the group D2d . Taking the superposition model, the
474
Y. Li et al. / Journal of Alloys and Compounds 484 (2009) 472–476 Table 2 Zero-field splitting parameters a, D and F for Cr2+ ions in CdGa2 S4 :Cr2+ system as a function of the three parameters R, , and N. N
R (Å)
(◦ )
a
D
F
0.985 0.985 0.985 0.965 0.965 0.965 0.945 0.945 0.945 0.925 0.925 0.925 Expt [11]
−0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03
2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70
0.042 0.049 0.061 0.042 0.048 0.059 0.042 0.048 0.58 0.043 0.048 0.057 0.043
−2.48 −2.54 −2.56 −2.39 −2.44 −2.46 −2.32 −2.36 −2.37 −2.25 −2.28 −2.29 −2.37
−0.042 −0.047 −0.056 −0.045 −0.049 −0.057 −0.047 −0.051 −0.060 −0.050 −0.054 −0.061 −0.054
3. Calculations for Cr2+ ions in the CdGa2 S4 :Cr2+ system 3.1. The local lattice structure distortion of Cr2+ ions in CdGa2 S4 Fig. 1. The local structure distortion of CdGa2 S4 :Cr system. R and denote the Cr–S bond length and angle between Cr–S bond and Z-axis. R0 and 0 are the Cd–S bond length and the angle between Cd–S bond and Z-axis of the host crystal CdGa2 S4 R and denote the structure distortion in CdGa2 S4 :Cr2+ system. 2+
s will vanish (the coordinates are chosen liangd-field parameters B44 as shown in Fig. 1) and the rest can be written as:
A2 (3 cos2 − 1) R3 1 A4 = (35 cos4 − 30 cos2 + 3) 2 R5 √ A4 = 70 5 sin4 R
B20 = 2 B40 c B44
where
A4 = −eq r 4 ,
(11)
The experimental ZFS data for the CdGa2 S4 :Cr2+ system have been reported [11]. From the experimental spectra, we can study the local geometric structure by diagonalizing the complete energy matrix. When Cr2+ doped in CdGa2 S4 host crystal, a Cr2+ will substitute for a Cd2+ ion, the surrounding local structure displays a tetragonal distortion, which can be described by two parameters R and (see Fig. 1). If one uses R0 and 0 to represent the Cd–S bond length and the angle between Cd–S bond and Z-axis of the host crystal CdGa2 S4 , respectively, then the local structure parameters R and for CdGa2 S4 :Cr2+ system may be expressed as R = R0 + R,
A2 = −eq r 2
(12)
R and denote the Cr–S bond length and angle between Cr–S bond and Z-axis, respectively, q the charge of ligand, and −e is the electron charge. The ratio r2 /r4 = 0.12205 a.u. can be calculated from the radial wave function [22]. A4 is almost a constant for the (CrS4 )6− tetrahedron, and can be determined from the absorption spectrum and the Cr–S bond length of the CrS crystal. For CrS, from its spectrum [23], we derive A4 = 57.6781 a.u. and A2 = 7.0395 a.u. for the (CrS4 )6− tetrahedron, and we will use them in the following calculations. Using the complete energy matrices and Eq. (11), the inter-relation between the local lattice structure parameters R and of CdGa2 S4 :Cr2+ system and ZFS parameters a, D and F can be established.
= 0 + ,
= 0 +
(13)
where R0 = 2.52 Å and 0 = 55.64◦ [24]. To decrease the number of adjustable parameters and reflect the covalency effects, we use the Curie et al.’s covalent theory and take an average covalence factor N to determine the optical parameters as following [25]: B = N 4 B0 ,
B = N 4 B0 ,
= N 2 0
(14)
where the free Cr2+ ion parameters [22] are B0 = 870.6 cm−1 , C0 = 3135.7 cm−1 and 0 = 232 cm−1 . For Cr2+ ions in CdGa2 S4 :Cr2+ system, by diagonalizing the complete energy matrices, the groundstate ZFS can be calculated with use of distortion parameters R and and covalence factor N. The comparisons between the theoretical values and experimental findings are shown in Table 2. It can be seen from Table 2 that the distortion parameters R = −0.06 Å and = 1.99◦ and the covalence factor N = 0.945 can provide a satisfactory explanation for the experimental ZFS parameters a, D and F. R < 0 indicates that the local lattice structure of
Table 3 Zero-field splitting parameters of CdGa2 S4 :Cr2+ system as a function of R, , and N. N
R (Å)
(◦ )
(a)a
(D)a
(F)a
(a)b
(D)b
(F)b
0.985 0.985 0.985 0.965 0.965 0.965 0.945 0.945 0.945 0.925 0.925 0.925
−0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03
2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70
0.036 0.043 0.055 0.035 0.041 0.052 0.034 0.040 0.049 0.033 0.038 0.047
−2.48 −2.54 −2.56 −2.39 −2.44 −2.46 −2.31 −2.35 −2.37 −2.25 −2.28 −2.29
−0.031 −0.036 −0.045 −0.032 −0.028 −0.022 −0.032 −0.037 −0.043 −0.033 −0.036 −0.043
0.007 0.010 0.015 0.006 0.008 0.013 0.005 0.007 0.011 0.004 0.006 0.009
−1.31 −1.37 −1.39 −1.21 −1.26 −1.28 −1.11 −1.16 −1.17 −1.02 −1.06 −1.07
−0.001 −0.002 −0.001 −0.001 −0.001 −0.005 −0.001 −0.002 −0.004 0 −0.001 −0.003
a b
Neglecting the spin singlets. Neglecting the spin singlets and the spin triplets.
Y. Li et al. / Journal of Alloys and Compounds 484 (2009) 472–476
475
Fig. 2. Spin singlets and spin triplets contributions to the fourth-order zero-field splitting parameters a and F of the CdGa2 S4 :Cr2+ system. Curve 1: neglecting the spin singlets; Curve 2: neglecting both the spin singlets and the spin triplets. Table 4 Spin singlets and spin triplets contributions to the zero-field splitting parameters a, D and F of CdGa2 S4 :Cr2+ system. N
R (Å)
(◦ )
(ra )a
(rD )a
(rF )a
(ra )b
(rD )b
(rF )b
0.985 0.985 0.985 0.965 0.965 0.965 0.945 0.945 0.945 0.925 0.925 0.925
−0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03 −0.09 −0.06 −0.03
2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70 2.20 1.99 1.70
0.1429 0.1224 0.0984 0.1667 0.1458 0.1186 0.1905 0.1667 0.1724 0.2326 0.2083 0.1715
0 0 0 0 0 0 0.0043 0.0042 0.0042 0 0 0
0.2619 0.2340 0.1964 0.2889 0.4288 0.6146 0.3191 0.2745 0.2833 0.3400 0.3333 0.2951
0.8333 0.7959 0.7541 0.8571 0.8333 0.7797 0.8810 0.8542 0.8103 0.9070 0.8750 0.8421
0.4718 0.4606 0.4570 0.4937 0.4836 0.4797 0.5216 0.5085 0.5063 0.5467 0.5351 0.5328
0.9762 0.9574 0.9821 0.9778 0.9796 0.9123 0.9787 0.9608 0.9333 1 0.9815 0.9508
a b
Neglecting the spin singlets. Neglecting the spin singlets and the spin triplets.
CdGa2 S4 :Cr2+ system has a compression distortion. The compression distortion may be ascribed to the fact that the radius of Cr2+ ion (r = 0.89 Å) is smaller than that of Cd2+ ion (r = 0.97 Å) [26]. From the calculation, the local lattice structure parameters R = 2.46 Å and = 57.63◦ for Cr2+ ions in CdGa2 S4 :Cr2+ system have been determined. Of course, careful experimental investigations, especially ENDOR experiments, are required in order to clarify the local lattice structure around the Cr2+ ions in CdGa2 S4 :Cr2+ system in detail. 3.2. The spin singlets contributions to ZFS of Cr2+ ions in CdGa2 S4 For further studying of the spin singlets contribution to the ZFS parameters of CdGa2 S4 :Cr2+ system, the ratios
a − a , a
ra =
D − D , D
rD =
F − F F
rF =
(15)
versus R, and N are calculated (Table 4), where a, D and F are the ZFS parameters considering all the spin states, and a , D and F those neglecting the spin singlets or neglecting both the spin singlets and spin triplets. ZFS parameters a , D and F are calculated can be seen in Table 3. The detailed results are presented in Figs. 2 and 3. It is obviously that the larger the ratio r, the larger are the contributions to the ZFS parameters. From Figs. 2 and 3, we can see that the spin singlets contributions to the ZFS parameter D are negligible, but the contributions to the ZFS parameters a and F cannot be neglected. It can be also seen from Figs. 2 and 3 that the values r for 5 D approximation is more than 0.46 for ZFS parameter D and 0.66 for ZFS parameter a and F. This means that the 5 D approximation method is imperfect. A part of the quantitative calculation results are listed in Tables 3 and 4.
Fig. 3. Spin singlets and spin triplets contributions to the second-order zero-field splitting parameter D of the CdGa2 S4 :Cr2 system. (Curve 1) Neglecting the spin singlets; (Curve 2) neglecting both the spin singlets and the spin triplets.
4. Conclusions A theoretical method for studying the inter-relation between molecular structure and electronic structure has been proposed by diagonalizing the complete energy matrices. From the above studies, we have the following conclusions: (i) The zero-field splitting parameters a, D and F of Cr2+ in CdGa2 S4 :Cr2+ system have been studied by the 210 × 210 complete energy matrix for a d4 configuration ion in tetragonal
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Y. Li et al. / Journal of Alloys and Compounds 484 (2009) 472–476
ligand-field. By diagonalizing the complete energy matrix, the local structure distortion parameters R = −0.06 Å and = 1.99◦ are determined. The results show that the local lattice structure of (CrS4 )6− coordination complex for Cr2+ ions in a CdGa2 S4 :Cr2+ system has an compression distortion. This distortion may be ascribed to the fact that the radius of Cr2+ ion (r = 0.89 Å) is smaller than that of Cd2+ ion (r = 0.97 Å). (ii) The spin singlets contributions to the zero-field splitting parameters of Cr2+ ions in CdGa2 S4 :Cr2+ system are investigated for the first time. It is shown that the spin singlets contributions to the zero-field splitting parameters a and F are important, and it cannot be neglected in general calculation. Acknowledgements This project was supported by National Natural Science Foundation of China (No. 10774103) and the Doctoral Education fund of Education Ministry of China (No. 20050610011). References [1] D. Haranath, H. Chander, N. Bhalla, P. Sharma, K.N. Sood, Appl. Phys. Lett. 86 (2005) 201904. [2] F.H. Su, Z.L. Fang, B.S. Ma, K. Ding, G.H. Li, W. Chen, J. Phys. Chem. B 107 (2003) 6991.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
J.H. Chung, C.S. Ah, D.J. Jang, J. Phys. Chem. B 105 (2001) 4128. A.A. Bol, A. Meijerink, J. Phys. Chem. B 105 (2001) 10197. A.A. Bol, A. Meijerink, J. Phys. Chem. B 105 (2001) 10203. Z. Quan, Z. Wang, P. Yang, J. Lin, J. Fang, Inorg. Chem. 46 (2007) 1354. P.A. Gonzalez Beermann, B.R. McGarvey, S. Muralidharan, R.C.W. Sung, Chem. Mater. 16 (2004) 915. Y.R. Do, Y.C. Kim, S.H. Cho, D.S. Zang, Y.D. Huh, S.J. Yun, Appl. Phys. Lett. 84 (2004) 1377. C.Z. Hadad, S.O. Vasquez, Phys. Rev. B 60 (1999) 8586. X.S. Jiang, W.R.L. Lambrecht, Phys. Rev. B 69 (2004) 035201. A.G. Avanesov, V.V. Badikov, G.S. Shakurov, Phys. Solid State 45 (2003) 1451. M.B. Johnson, S.B. Mirov, V. Fedorov, M.E. Zvanut, J.G. Harrison, V.V. Badikov, G.S. Shevirdyaeva, Opt. Commun. 233 (2004) 403. L. Deloach, R. Page, G. Wilke, S. Payne, W. Krupke, IEEE J. Quant. Electron. 32 (1996) 885. C. Rudowicz, Y.Y. Zhou, L.W. Yu, J. Phys. Chem. Solids 53 (1992) 1227. C. Rudowicz, Y.Y. Zhou, J. Magn. Magn. Mater. 111 (1992) 153. C. Rudowicz, M.L. Du, Y.Y. Yeung, Y.Y. Zhou, Physica B 191 (1993) 323. Y.Y. Zhou, C.L. Li, Phys. Rev. B 48 (1993) 16489. Y.Y. Zhou, Phys. Rev. B 50 (1994) 7161. Z.F. Li, D.H. Li, Y.Y. Zhou, Physica B 252 (1998) 167. U. Kaufmann, Phys. Rev. B 14 (1976) 1848. D.J. Newman, W. Urban, Adv. Phys. 24 (1975) 793. M.G. Zhao, L.H. Xie, Mater. Sci. Eng. B 75 (2000) 72. T.C. De Vore, Ph.D. Thesis, Iowa State University, USA 1975. A.A. Lavrentyev, B.V. Gabrelian, V.A. Dubeiko, I.Ya. Nikiforov, J.J. Rehr, J. Phys. Chem. Solids 63 (2002) 227. D. Curie, C. Barthon, B. Canny, J. Chem. Phys. 61 (1974) 3048. Z.M. Zhou, Y. Xu, Z.M. Wang, H.D. Jia, Concise Inorganic Chemistry, Zhengzhou University Press, China, 2002.