Theoretical study on local lattice structure distortion for octahedral Fe3+ center in several germanate garnets

Chemical Physics Letters 441 (2007) 143–147 www.elsevier.com/locate/cplett

Theoretical study on local lattice structure distortion for octahedral Fe3+ center in several germanate garnets Cai-Xia Zhang a, Xiao-Yu Kuang a b

a,b,*

, Guang-Dong Li a, Hui Wang

a

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China International Centre for Materials Physics, Academia Sinica, Shenyang 110016, China Received 31 December 2006; in final form 9 April 2007 Available online 29 April 2007

Abstract Local lattice structure distortion study of octahedral Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) has been performed systematically based on the diagonalization of the complete energy matrices of the electron–electron repulsion, the spin–orbit coupling and the trigonal ligand–field interaction for a d5 configuration ion in a trigonal ligand field. From the EPR calculation, the local structure distortion parameters DR and Dh are determined, respectively. These results illuminate a microscopic origin of various ligand field parameters used empirically for the interpretation of electron paramagnetic resonance experiments. It is found that the theoretical results of the electron paramagnetic resonance spectra for Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) are in good agreement with the experimental findings. Although the local lattice structures around the M3+ (M = Al, Ga, Sc, In, Lu) ions are obviously different, the local lattice structures around the dopants are rather similar. Moreover, it is noted that the empirical impurity–ligand distance R  RH þ 12 ðri  rh Þ is not suitable for Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu).  2007 Elsevier B.V. All rights reserved.

1. Introduction Garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) doped with transition metal ions (Ni2+, Mn2+, Fe3+, etc.) or rare earth metal ions (Er3+, Yb3+, Nd3+, etc.) have attracted intense interest due to their very important optical properties for wide use in lasers operating in the visible, near infrared, infrared spectral ranges and optical communications [1–8]. In order to understand the influence of these impurities on the optical properties, a great deal of effort has been devoted to studying it during the recent several decades [9–13], especially, P. Nova´k et al. had performed the EPR spectra of the octahedral Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) [9]. Their experimental results give important information about the ground state of the transition metal Fe3+ ions and form a * Corresponding author. Address: Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China. E-mail addresses: [email protected] (C.-X. Zhang), scu_kxy@ 163.com (X.-Y. Kuang).

0009-2614/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.04.087

useful starting point for understanding the inter-relationships between electronic and molecular structure of Fe3+ ions in the (FeO6)9 coordination complex, from which we can study the characteristics of the local lattice structure around the octahedral Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu). Electron paramagnetic resonance (EPR) is regarded as an effective tool to study the microstructure and the local environment around a substitution magnetic ion site in crystals [14,15]. The reason is that the electron paramagnetic resonance spectra usually provide highly detailed microscopic information about the structure of the defects. In fact, a careful investigation of local lattice structure around the dopants is an essential step for a complete understanding of the optical properties. The local lattice structure around the Fe3+ in the crystals has trigonal symmetry about the crystalline c-axis. For a d5 configuration ion in a trigonal ligand field, the high-spin ground-state is the 6A1 state. To describe the 6 A1 ground-state splitting of the octahedral Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu), the spin

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Hamiltonian should include three different EPR zero-field splitting parameters B02 , B04 and B34 . It is known that the EPR parameters B02 , B04 are associated with the secondand fourth-order spin operators and represent an axial component of the crystalline electric field, respectively. The parameter B34 relates to a fourth-order spin operator and represents a cubic component of the crystalline electric field. Consequently, this may help us to get valuable information about the local lattice structure distortion. In the present Letter, the correlation between EPR parameters and local lattice structure distortion of octahedral Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) will be established and investigated by diagonalizing the complete energy matrices for electron–electron repulsion, spin–orbit coupling and trigonal ligand–field interaction. The distortion parameters DR and Dh for the octahedral Fe3+center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) are determined, important characteristics of the local lattice structure distortion around the octahedral Fe3+center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) are revealed, the EPR parameters B02 , B04 , B34 are also reasonably interpreted. It is shown that the empirical formula of impurity–ligand distance R  RH þ 12 ðri  rh Þ is not suitable for Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) [16]. 2. Theoretical methods 2.1. Method for the construction of the complete energy matrix of the d5 configuration ion in trigonal ligand field The method of diagonalizing the complete energy matrices has proved to be a powerful tool in the study of EPR zero-field splitting parameters and the local lattice structure [17,18]. For a d5 configuration ion in a trigonal ligand field, in order to construct the complete energy matrices, the jL, S, ML, MSæi basic functions need to be expanded into the form of the Ui basic function X C j Uj ð1Þ jL; S; M L ; M S ii ¼ j

where Cj is the Clebsch–Gordon coefficient, and Uj is one of the 252 basic Slater determinants. The basic functions for the d5 configuration ion can be obtained in terms of the L± and S± operators from the jL, S, ML, MSæi functions with ML = 0 and MS = 0. By employing the jL, S, ML, MSæi functions of the d5 configuration ion in trigonal symmetry, we have constructed the complete energy matrix (252 · 252) of Hamiltonian [19] b ¼H b ee þ H b so þ H b lf ¼ H

X e2 X X þf li s i þ V i; ri;j i
ð2Þ

b ee , H b so and H b lf represent, respectively, the elecwhere H tron–electron interaction, the spin-orbit coupling interaction and the ligand-field potentials that may be expressed as

b lf ¼ R c00 Z 00 þ c20 r2 Z 20 ðhi ; ui Þ þ c40 r4 Z 40 ðhi ; ui Þ H i i i

þ cc43 r4i Z c43 ðhi ; ui Þ þ cs43 r4i Z s43 ðhi ; ui Þ;

ð3Þ

where ri, hi and ui are spherical coordinates of the ith electron. Zlm, Z clm and Z slm are respectively, defined as Z l0 ¼ Y l0 ; pffiffiffi m Z clm ¼ ð1= 2Þ½Y l;m þ ð1Þ Y l;m ; pffiffiffi Z slm ¼ ði= 2Þ½Y l;m  ð1Þm Y l;m :

ð4Þ

The Yl,m in Eq. (4) are the spherical harmonics. cl0, cclm and cslm are associated with the local lattice structure around d5 configuration ion by the relations cl0 ¼ 

n 4p X eqs Z l0 ðhs ; us Þ; 2l þ 1 s¼1 Rlþ1 s

cclm ¼ 

n 4p X eqs c Z lm ðhs ; us Þ; 2l þ 1 s¼1 Rlþ1 s

cslm ¼ 

n 4p X eqs s Z lm ðhs ; us Þ; 2l þ 1 s¼1 Rlþ1 s

ð5Þ

where hs and us are angular coordinates of the ligand. s and qs represent the sth ligand ion and its effective charge, respectively. Rs denotes the impurity–ligand distance. The 252 · 252 energy matrix for a d5 configuration ion corresponding to the Hamiltonian (2) have been derived [19]. The matrix elements are the functions of the Racah parameters B and C, Trees correction a, seniority correction b, the spin–orbit coupling coefficient f, and the ligand-field parameters B20 ; B40; Bc43 , which are of the forms for Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) as [20]: 1X G2 ðsÞ ð3 cos2 hs  1Þ; 2 s 1X B40 ¼ G4 ðsÞ ð35 cos4 hs  30 cos2 hs þ 3Þ; 8 s pffiffiffiffiffi 35 X c G4 ðsÞ ðsin3 hs cos hs cos 3/s Þ; B43 ¼ 4 s

B20 ¼

ð6Þ

where G2(s) and G4(s) are represented as G2 ðsÞ ¼ eqs G2 ðsÞ; G4 ðsÞ ¼ eqs G4 ðsÞ; Z Rs Z 1 k Rks k 2 2 r G ðsÞ ¼ R3d ðrÞr kþ1 dr þ R23d ðrÞr2 kþ1 dr: r Rs 0 Rs

ð7Þ ð8Þ

In Eqs. (6) and (8), the Rs and hs denote the Fe–O distance and the angle between Fe–O bond and C3 axis, respectively. R3d(r) is the radial wavefunction of the 3d-electron. According to the Van Vleck approximation for Gk(s) integral [21], we have the following relations

C.-X. Zhang et al. / Chemical Physics Letters 441 (2007) 143–147

A2 R3 A4 G4 ðsÞ ¼ 5 ; R where G2 ðsÞ ¼

ð9Þ

A4 ¼ eqs hr4 i; A2 =A4 ¼ hr2 i=hr4 i: As far as we obtain the value of A4 from the corresponding optical spectra and estimate the ratio of Ær2æ/Ær4æ from radial wave functions, then we can conveniently get the value of A2. Based on the diagonalization of the complete energy matrices, we can calculate the ground-state zerofield splitting of Fe3+ ion in (FeO6)9 cluster and determine the distortion of the local lattice structure from EPR spectra. 2.2. The EPR zero-field splitting parameters in a trigonal ligand-field For Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu), the EPR spectra of d5 configuration Fe3+ ion in a trigonal ligand field can be analyzed by employing the spin Hamiltonian [22] *

*

The positive sign in this equation corresponds to B02 P 0 and negative otherwise, respectively. 3. Calculations

A2 ¼ eqs hr2 i;

b ¼ gb H  S þB0 O0 þ B0 O0 þ B3 O3 : H 2 2 4 4 4 4

145

ð10Þ

Where Omn are the standard Stevens spin operators and Bml are the zero-field splitting EPR parameters. Solving the spin Hamiltonian (10) within the basis of states j S ¼ 5 ; M S i, the splitting energy levels in the ground-state 6A1 2 for a zero magnetic field can be written as 1   1 ½ð54B02  180B04 Þ2 þ 3240ðB34 Þ2 2 0 0 : E  ¼ B2 þ 90B4  2 6   3 E  ¼ 2B02  180B04 : 2 1   5 ½ð54B02  180B04 Þ2 þ 3240ðB34 Þ2 2 : E  ¼ B02 þ 90B04  2 6

When Fe3+ impurities are doped in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu), a number of Fe3+ ions will substitute for M3+ (M = Al, Ga, Sc, In, Lu) ions in the octahedral sublattice [9] and the local lattice structure around the octahedral Fe3+ center displays a trigonal distortion. The distortion can be described by employing two parameters DR and Dh as shown in Fig. 1. If we use R0, h0 to represent the M–O bond length and the angle between M–O bond and C3 axis of the host crystal Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu), respectively, then the local lattice structure parameters R and h for the octahedral Fe3+ center may be expressed as R ¼ R0 þ DR; h ¼ h0 þ Dh:

ð13Þ

As long as we know the values of A4 and A2, the trigonal ligand-field parameters B20, B40 and Bc43 are only the functions of DR and Dh. In order to reduce the number of adjustable parameters and to reflect the effects of covalency, we take the typical average covalence factor N = 0.91 as found in the MgO:Fe3+ system for Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) [23] and use the following relationship: B ¼ N 4 B0 ;

C ¼ N 4C0;

1 1 DE1 ¼  ½ð180B04 þ 54B02 Þ2 þ 3240ðB34 Þ2 2 : 3 1 2 2 1 DE2 ¼ 270B04  3B02  ½ð180B04 þ 54B02 Þ þ 3240ðB34 Þ 2 : 6 ð12Þ

b ¼ N 4 b0 ;

f ¼ N 2 f0 ; ð14Þ

1

where B0 = 1106 cm , C0 = 3922 cm , a0 = 81 cm1, b0 =  29 cm1, f0 = 470 cm1 are the free Fe3+ ion parameters [24,25]. The ratio of Ær2æ/Ær4æ = 0.097 is obtained from the radial wave function of Fe3+ ion in complexes [26]. A4 is almost a constant for octahedral (FeO6)9 cluster, its value can be determined from the

1

C3 ΔR

Fe3+

ð11Þ   E 32 corresponds   to the irreducible representation C6, E  12 and E  52 correspond to C4 or C5. The 6A1 ground-state zero-field splitting energies DE1, DE2 may be expressed as

a ¼ N 4 a0 ;

R0

θ0

θ

O 2−

R Δθ

M 3+

O 2−

Fig. 1. Local structure distortion of octahedral Fe3+ centre in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu). R0, h0 are the structure parameters of host crystals. R, h are the structure parameters when Fe3+ replaces M3+ (M = Al, Ga, Sc, In, Lu). DR, Dh denote the structure distortion.

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C.-X. Zhang et al. / Chemical Physics Letters 441 (2007) 143–147

optical spectra and Fe–O bond length of the aFe2O3 crystal [27,28]. A4 = 27.6967 au and A2 = 2.6870 au are derived 9 for octahedral ðFeO6 Þ cluster. For Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu), the ground-state zero-field splitting may be simulated with use of the distortion parameters DR and Dh by diagonalizing the complete energy matrix. Simultaneously, the distortion parameters DR and Dh are determined, respectively. The comparison between the theoretical values and the experimental findings are collected in Table 1. It can be seen from Tables 1 and 2. that the distortion parameters DR and Dh can provide an excellent explanation of the experimental findings of EPR parameters. It is shown that the local lattice structures around the octahedral Fe3+ centers in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) have obvious distortions, which may be attribTable 1 The ground-state zero-field splittingDE1, DE2 and the EPR parameters B34 , B02 , B04 for octahedral Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu ) as a function of DR and Dh (T = 77 K) M

Al

Dh

104DE1

104DE2

104 B02

104 B04

104 B34

0.1181 0.1197 0.1190 0.1173 0.1190

0.230 0.240 0.230 0.213 0.223

2453.3 2454.5 2440.9 2425.4 2424.4

627.7 631.7 625.5 617.1 620.6

135.8 135.9 135.1 134.3 134.2 135.1

0.710 0.696 0.703 0.714 0.700 0.704

26.2 26.2 26.2 26.2 26.2 26.2

0.0608 0.0581 0.0590 0.0573 0.0610

0.051 0.036 0.035 0.018 0.040

2158.7 2158.6 2145.9 2129.7 2133.0

554.6 550.2 547.8 539.3 547.1

119.3 119.4 118.6 117.7 117.8 118.6

0.618 0.634 0.627 0.638 0.615 0.626

23.9 23.9 23.9 23.9 23.9 23.9

0.0336 0.0366 0.0356 0.0373 0.0332

0.023 0.040 0.041 0.058 0.034

2275.2 2276.7 2262.7 2248.9 2248.8

617.9 614.0 610.7 604.3 609.7

126.3 126.4 125.6 124.8 124.8 125.5

0.522 0.538 0.533 0.540 0.520 0.532

20.5 20.5 20.5 20.5 20.5 20.5

0.0698 0.0728 0.0718 0.0734 0.0694

0.289 0.305 0.307 0.322 0.301

1714.0 1709.7 1697.1 1681.8 1687.1

437.5 432.3 429.7 422.0 429.1

94.7 94.6 93.8 93.0 93.2 93.9

0.500 0.515 0.508 0.517 0.499 0.507

19.0 19.0 19.0 19.0 19.0 19.0

0.1220 0.1259 0.1249 0.1264 0.1225

0.528 0.606 0.606 0.624 0.600

2219.3 2218.4 2206.9 2189.9 2191.6

613.9 608.8 606.5 599.0 604.3

123.2 123.2 122.5 121.6 121.6 122.4

0.468 0.485 0.479 0.486 0.469 0.478

19.0 19.0 19.0 19.0 19.0 19.0

DR

Expt. [9]

Ga

Expt. [9]

Sc

Expt. [9]

In

Expt. [9]

Lu

Expt. [9]

DR is in unit of A, Dh is in unit of deg, 104DE1, 104DE2, 104 B02 , 104 B04 and 104 B34 are in units of cm1. The estimated uncertainties of experimental findings 104 B34 , 104 B02 and 104 B04 are ±0.5 to ± 1.0 cm1, ±0.5 to ±0.8 cm1, and ±0.005 to ±0.008 cm1, respectively [9].

Table 2 The interrelation between the local lattice parameters R, h and lattice constant a (A), ion radius ri (A) of M3+ ion in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu ) ( T = 77 K) M

a (A)

ri (A)

R0 (A)

h0 (deg)

DR (A)

Dh (deg)

R (A)

h (deg)

Al Ga Sc In Lu

12.120 12.260 12.519 12.590 12.735

0.51 0.62 0.73 0.81 0.85

1.92 1.99 2.10 2.14 2.20

53.90 53.78 53.56 53.56 52.96

0.1190 0.0590 0.0356 0.0718 0.1249

0.230 0.035 0.041 0.307 0.606

2.0390 2.0490 2.0644 2.0682 2.0751

53.670 53.745 53.601 53.867 53.566

The estimated uncertainties of experimental findings R0 (A) and h0 (deg) are ±0.01 (A) and 0.50 (deg), respectively [30].

uted to the fact that the radius of Fe3+ (r = 0.64 A) ion is different from that of M3+ (M = Al, Ga, Sc, In, Lu) ions [29]. We find that the distortion parameters DR > 0, Dh < 0 for Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga), this means that the local lattice structure around the octahedral Fe3+ center has an elongation distortion and the distortion feature may be ascribed to the fact that the radius of Fe3+ (r = 0.64 A) ion is larger than that of Al3+ (r = 0.51 A) ion or Ga3+ (r = 0.62 A) ion. Whereas, for Fe3+ in garnets Ca3M2Ge3O12 (M = Sc, In, Lu), the local lattice structure around the octahedral Fe3+ center has another distortion characteristics, i.e., DR < 0 and Dh > 0, which may be ascribed to the fact that the radius of Fe3+ (r = 0.64 A) ion is smaller than that of Sc3+ (r = 0.73 A) ion, In3+ (r = 0.81 A) ion or Lu3+(r = 0.85A) ion. Meanwhile, the local lattice structure parameters R and h for octahedral Fe3+ ion center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) are determined. From Table 2, we can also find that in spite of the local lattice structures around the octahedral M3+ centers in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) are obviously different, for example, the range of R0 = 1.92 A for M = Al to R0 = 2.20 A for M = Lu [30], however, when M3+ (M = Al, Ga, Sc, In, Lu) ions are replaced by Fe3+ions, the local lattice structures around the octahedral Fe3+ centers in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) are rather similar, the range of Fe–O bond lengths are 2.039 A 6 R 6 2.075 A and the range of angles between Fe–O bond and C3-axis are 53.566 6 h 6 53.867. We note that the larger the lattice constant a (A) in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu), the larger the impurity–ligand distance, while the bond angles for the octahedral Fe3+ center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) are almost the same. This physical picture may help us to understand the optical properties of the materials from technological and applicable viewpoints. It is also worthwhile to note that the approximate formula about R  RH þ 12 ðri  rh Þ (whereRH is the cation–ligand distance corresponding to the perfect host lattice, ri and rh are the ionic radii of impurity and host cation, respectively) not suitable for Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu). For instance, for Fe3+ in Ca3Al2Ge3O12, the estimated Fe–O distance is R  1.985 A (for

C.-X. Zhang et al. / Chemical Physics Letters 441 (2007) 143–147

Al–O distance, RH = 1.92 A; for Fe3+, ri = 0.64 A and for Al3+, rh = 0.51 A), this result is obviously less than R = 2.039 A determined from the EPR calculation. Of course, more experiments, especially electron nuclear double resonance (ENDOR) experiments are needed to elucidate our results. 4. Conclusions The local lattice structures of octahedral Fe3+ centers in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu) have been studied. It is demonstrated that there is an obvious distortion for the octahedral Fe3+ ion center in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu). The local lattice structure distortion parameters (DR, Dh) for the octahedral Fe3+ center are determined. Our results indicate that DR > 0, Dh < 0 for Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga) and DR < 0, Dh > 0 for Fe3+in garnets Ca3M2Ge3O12 (M = Sc, In, Lu). A significant result is reported that in spite of the local lattice structures around the M3+ (M = Al, Ga, Sc, In, Lu) ions are obviously different, the local lattice structures around the dopants are rather similar. In addition, we confirm that the formula for the impurity–ligand distance R  RH þ 12 ðri  rh Þ is not suitable for Fe3+ in garnets Ca3M2Ge3O12 (M = Al, Ga, Sc, In, Lu). Acknowledgements This work was supported by the National Natural Science Foundation of China (No.10374068) and the Doctoral Education Fund of Education Ministry of China (No. 20050610011). References [1] L.H. Huang, X.R. Liu, W. Xu, B.J. Chen, J.L. Lin, J. Appl. Phys. 90 (2001) 5550.

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