EPR theoretical study of local lattice structure in Al2O3:Fe3+ system

Journal of Physics and Chemistry of Solids 65 (2004) 1147–1151 www.elsevier.com/locate/jpcs

EPR theoretical study of local lattice structure in Al2O3:Fe3þ system Wei Lua, Xiao-Yu Kuanga,b,*, Kang-Wei Zhoua,b, Dong Diec a

Department of physics, Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China b International Centre for Materials Physics, Academia Sinica, Shengyang 110016, China c Department of Electronic Information and Electrical Engineering, Xihua University, Chengdu 610039, China Received 4 June 2003; accepted 6 January 2004; available online 5 March 2004

Abstract By analyzing the EPR spectrum of transition-metal ion Fe3þ in Al2O3:Fe3þ system, the local lattice structure around impurity Fe3þ ion in the crystal has been studied by means of the diagonalization of the energy matrices of the electron – electron repulsion, the ligand-field and the spin– orbit coupling for a d5 configuration ion in a trigonal ligand-field. Both the second-order and fourth-order EPR parameters D and ða 2 FÞ are taken simultaneously in the structural investigation. The results indicate that the two three-edge-pyramids elongated obviously along C3 axis. The two distortion angles Du1 ¼ 21:1 ^ 0:18; Du2 ¼ 21:88 as well as the two Fe – O bond lengths R1 ¼ 2:016 A, R2 ¼ 1:907 A are determined, respectively. q 2004 Elsevier Ltd. All rights reserved. Keyword: D. Crystal structure

1. Introduction Corundum (Al2O3) doped with transition-metal ions (Fe, Cr, Ti, Cu, etc.) has been proved to be an important ceramic and laser crystal material [1 – 9]. The micro-structural distortion of local lattice where transition-metal ion locates in the crystal has been discussed in many works and different viewpoints have been proposed. For instance, as for Al2O3:Mnþ system, McClure has suggested that the transition-metal ion Mnþ does not occupy the Al3þ ion site exactly, but is displaced from it along threefold axis towards the empty octahedral site [6]. According to this assumption, recently, Zheng has studied the local structure of Al2O3:Fe3þ based on the EPR parameter D [7]. His conclusion is that the Fe3þ will shift to the bigger oxygen triangle when Fe3þ replaces Al3þ in Al2O3:Fe3þ system. However, it is well known that for a d5 configuration ion in a trigonal ligand-field, the high-spin ground state is the 6A1 state. To reproduce the 6A1 ground state splitting of Fe3þ in Al2O3:Fe3þ, generally, the spin-Hamiltonian includes three * Corresponding author. Address: Department of physics, Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China. E-mail address: [email protected] (X.-Y. Kuang). 0022-3697/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2004.01.003

different EPR parameters a; D and ða 2 FÞ [10]. Parameter a is associated with a fourth-order spin operator and represents a cubic component of the crystalline electric field. Parameters D and ða 2 FÞ are, respectively, the second-order and fourth-order spin operators, and represent a component of the crystalline electric field that is axially symmetric about the C3 axis. Since both the EPR parameters D and ða 2 FÞ relate to the axial ligand field, herein, we will suggest that the two low-symmetry EPR parameters D and ða 2 FÞ should be simultaneously considered in the determination of the distortion of the local crystal structure of Fe3þ in Al2O3:Fe3þ. Because the radius of the transitionmetal ion Fe3þ is obviously larger than that of Al3þ ion [11], we may predict that when Fe3þ replaces the host ion Al3þ in Al2O3, the Fe3þ will push the two oxygen triangles to move along C3 axis and this will lead to an elongate effect of the sub-lattice. By diagonalizing the energy matrices of the electron – electron repulsion, the ligand-field and the spin – orbit coupling for a d5 configuration ion in a trigonal ligandfield [4], the relationship between the EPR parameters D; ða 2 FÞ and the distortion of local crystal structure will be studied and both the bond lengths and bond angles in Al2O3:Fe3þ will be determined by simulating the two lowsymmetry EPR parameters D and ða 2 FÞ simultaneously.

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2. Energy matrix and EPR parameter The perturbation Hamiltonian for a d5 configuration ion in a trigonal ligand-field may be represented as [4]: X 2 X X ^ ee þ H^ so þ H^ CF ¼ H^ ¼ H e =ri;j þ z li si þ Vi ; ð1Þ i,j

i

i

where z is the spin – orbit coupling coefficient and Vi is the ligand-field potential Vi ¼ g00 Z00 þ g20 ri2 Z20 ðui ; wi Þ þ g40 ri4 Z40 ðui ; wi Þ c s þgc43 ri4 Z43 ðui ; wi Þ þ gs43 ri4 Z43 ðui ; wi Þ:

ð2Þ

On the basis of the irreducible representations G4 ðG5 Þ and G6 of C3* point group, for a d5 configuration ion three 84 £ 84 energy matrices corresponding to the perturbation Hamiltonian (1) have been derived [4]. The matrix elements are the functions of the Racah parameters B and C; the spin – orbit coupling coefficient z; and the ligand field parameters that are of the forms:   5 1=2 B20 ¼ g20 , r2 .; 4p   1 1=2 g40 , r4 .; B40 ¼ 3 4p   ð3Þ 3 1 1=2 c g43 , r4 .; Bc43 ¼ 2 2p   3 1 1=2 s g43 , r4 . : Bs43 ¼ i 2 2p For Fe3þ in Al2O3:Fe3þ, the local structure is composed of two asymmetry three-edge-pyramids and the Fe3þ ion is surrounded by six oxygen ions [6]. The local symmetry may be approximated as C3v (See Fig. 1). We use p1 ; p2 ; respectively, to represent the ligand ions in the up and down pyramids in Al2O3:Fe3þ and use u1 ; u2 to represent the corresponding angles between metal-ligand bonds and C3 axis. In this case the explicit expressions of Bkq may be written as: B20 B40

Bc43

3 ¼ ½G2 ðp1 Þð3cos2 u1 2 1Þ þ G2 ðp2 Þð3cos2 u2 2 1Þ; 2 3 ¼ ½G4 ðp1 Þð35cos4 u1 2 30cos2 u1 þ 3Þ 8

Gk ðpi Þ ¼

ð R pi 0

R23d ðrÞr 2

ð4Þ

ð5Þ k ð1 rk 2 2 Rpi dr þ R ðrÞr dr: 3d Rkþ1 r kþ1 Rpi pi

G2 ðp1 Þ – G2 ðp2 Þ;

ð6Þ

ð7Þ

G4 ðp1 Þ – G4 ðp2 Þ: According to the Van Vleck approximation for Gk ðpi Þ integral and using the point charge model [13], we have the relations

G4 ðp2 Þ ¼ ðRp1 =Rp2 Þ5 G4 ðp1 Þ;

where G2 ðpi Þ and G4 ðpi Þ are expressed as: G4 ðpi Þ ¼ qeG4 ðpi Þ;

The ratio of G2 =G4 which depends on the bond lengths has been studied by Gerloch and Slade [12]. By using the Gerloch et al’s method and considering the bond lengths in Al2O3:Fe3þ, we estimate G2 =G4 ¼ 1:5 for Al2O3:Fe3þ system. The ratio G2 =G4 ¼ 1:5 will be taken in our following calculations. Since the bond lengths in the two pyramids in Al2O3:Fe3þ are not same, we may predict that

G2 ðp2 Þ ¼ ðRp1 =Rp2 Þ3 G2 ðp1 Þ;

þ G4 ðp2 Þð35cos4 u2 2 30cos2 u2 þ 3Þ; 3 pffiffiffi 35½G4 ðp1 Þcosu1 sin3 u1 þ G4 ðp2 Þcosu2 sin3 u2  ¼ 4

G2 ðpi Þ ¼ qeG2 ðpi Þ;

Fig. 1. Local structure of Al2O3:Fe3þ system.

ð8Þ

where G4 ðp1 Þ ¼

2G4 ; 1 þ ðR1 =R2 Þ5

G2 ðp1 Þ ¼

2G2 : 1 þ ðR1 =R2 Þ3

With use of the expressions (4) and (8), the optical parameters B; C; z and G4 of Fe3þ in Al2O3:Fe3þ and the energy matrices [4], we can calculate the ground state zerofield splitting as well as the EPR parameters D and ða 2 FÞ as the functions of distortion angles in Al2O3:Fe3þ.

W. Lu et al. / Journal of Physics and Chemistry of Solids 65 (2004) 1147–1151

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The EPR spectra of Fe3þ ion in a trigonal ligand-field may be analyzed by employing the spin-Hamiltonian [10]: ^ s ¼ gbH^ S^ þ D½S2z 2 SðS þ 1Þ=3 H þ a½S4j þ S4h þ S4z 2 SðS þ 1Þð3S2 þ 3S 2 1Þ=5=6 þ F½35S4z 2 30SðS þ 1ÞS2z þ 25S2z 2 6SðS þ 1Þ þ 3S2 ð2S þ 1Þ2 =180:

ð9Þ

From (9) the explicit expression of the energy in 6A1 state for a zero magnetic field can be written as:   1 E ^ ¼ D=3 2 ða 2 FÞ=2 2 ½ð18D þ a 2 FÞ2 þ 80a2 1=2 =6; 2   3 ¼ 22D=3 þ ða 2 FÞ; E ^ 2   5 E ^ ¼ D=3 2 ða 2 FÞ=2 þ ½ð18D þ a 2 FÞ2 þ 80a2 1=2 =6: 2 ð10Þ Eð^ð3=2ÞÞ corresponds to the irreducible representation G6 ; Eð^ð1=2ÞÞ and E ^ ð5=2Þ to G4 (or G5 Þ: The 6A1 ground state zero-field splitting energies DE1 ; DE2 may be expressed as: 1 DE1 ¼ ^ ½ða 2 F þ 18DÞ þ 80a2 1=2 ; 3 3 1 DE2 ¼ ða 2 FÞ 2 D ^ ½ða 2 F þ 18DÞ2 þ 80a2 1=2 ; 2 6 ð11Þ where the positive and negative signs in Eq. (11) correspond to D $ 0 and D , 0; respectively, [14]. For Al2O3:Fe3þ system, it was demonstrated that the low-symmetry EPR parameters D and ða 2 FÞ are almost independent on the EPR cubic parameter a [4]. Therefore, we can use a cubic approximation to calculate the EPR parameter a [4,15] and fix it when we study the relationship between the lowsymmetry EPR parameters D; ða 2 FÞ and the trigonal distortion in Al2O3:Fe3þ system.

3. Displacement model The local crystalline structure of Al2O3 involves two asymmetrical three-edge-pyramids which are situated at up and down parts of the crystal sub-lattice along threefold axis, respectively. When transition-metal ions dope in Al2O3, McClure suggested that the transition-metal ion does not occupy the Al3þ ion site exactly, but is displaced from it along the threefold axis towards the empty octahedral site [6]. By taking z-axis along C3 axis of Al2O3, the local distortion may be described by use of a displacement DZ as plotted in Fig. 2. DZ . 0 and DZ , 0 represent the shift of Fe3þ towards up and down oxygen triangles, respectively. This is the displacement model.

Fig. 2. Displacement Model of Fe3þ in Al2O3:Fe3þ system. R10 ¼ 1.966 A, R20 ¼ 1.857 A, u10 ¼ 47:648; u20 ¼ 63:068; L10 ¼ 1.452 A, L20 ¼ 1.653 A; R10 ; R20 are the structure parameters of host crystal Al2O3; R1 ; R2 ; u1 and u2 are the structure parameters when Fe3þ replaces Al3þ, DZ represents the shift along threefold axis.

If one uses R10 ; R20 ; u10 and u20 to represent the Al– O bond lengths and the angles between Al – O bonds and C3 axis in the up and down pyramids in host crystal Al2 O 3, respectively, then the local structure parameters R1 ; R2 ; u1 and u2 for Fe3þ replacing Al3þ in Al2O3: Fe3þ system can be expressed as R1 ¼ ½ðR10 sinu10 Þ2 þ ðR10 cosu10 2 DZÞ2 1=2 ; R2 ¼ ½ðR20 sinu20 Þ2 þ ðR20 cosu20 þ DZÞ2 1=2 ;   L10 u1 ¼ tg21 ; R cosu10 2 DZ  10  L20 u2 ¼ tg21 ; R20 cosu20 þ DZ

ð12Þ

ð13Þ

where L10 and L20 are the distances between O2- and threefold axis in the up and down oxygen triangles, respectively [6]. From (13) we can see that in the displacement model the shift DZ will lead to a reverse change of the angles u1 and u2 ; i.e. when one is increasing, the other must be decreasing. By taking the optical parameters B ¼ 660 cm21, C ¼ 3135 cm21, z ¼ 360 cm21 [5] and G4 ¼ 9060 cm21. We calculate the EPR secondorder and fourth-order parameters D and ða 2 FÞ vs. the displacement DZ by diagonalizing the energy matrices of the electron – electron repulsion, the ligand-field and the spin – orbit coupling interaction for a d5 configuration ion in a trigonal ligand-field. As shown in Table 1,

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Table 1 The ground-state splittings DE1 ; DE2 and the EPR zero-field splitting parameters a; D and ða 2 FÞ for Fe3þ ion in Al2O3:Fe3þ as a function of the displacement DZ in the displacement model, where 104 DE1 ; 104 DE2 ; 104 D; 104 ða 2 FÞ; are in units of cm21 ; 104 a ¼ 237 cm21 for Dq ¼1536 cm21

u1 ¼ u10 þ Du1 ;

104 DE2

104 D

104 ða 2 FÞ

104 a

7270 8382 9492

2658 3019 3379

1198 1384 1570

147 141 136

237 237 237

20.0163

10382

3669

1719

131

237

Du1 ¼ sin21 ðL10 =R1 Þ 2 u10 ¼ 21:68;

20.02 20.04

10581 11632

3733 4074

1752 1928

130 124

237 237

Du2 ¼ sin21 ðL20 =R2 Þ 2 u20 ¼ 22:88:

Expt. [16]

10451

4014

1719

339

236

DZ (A) 0.04 0.02 0

104 DE1

to evaluate the Fe – O bond lengths in Al2O3:Fe3þ system. If one assumes the O – O bond length in oxygen triangles does not change (rigid condition), then the limit distortion angles Du1 and Du2 can be derived by using the geometric relation. We have

the calculated EPR second-order parameter 104 Dcalc: ¼ 1719 cm21 for DZ ¼ 20:0163 A can agree well with the experimental finding 104 Dexpt: ¼ 1719 cm21 [16]. Here, DZ , 0 means that the Fe3þ ion moves towards the bigger oxygen triangle. This conclusion is same as that reported by Zheng [17]. His calculation is based on the superposition model [18]. However, we also note that from the displacement model the calculated EPR fourth-order parameter ða 2 FÞ always deviates far from the experimental datum. The theoretical values of 104 ða 2 FÞcalc: ¼ 124 , 147 cm21 are found to be too small by a factor 2 – 3 of the experimental finding 104 ða 2 FÞexpt: ¼ 339 cm21 and this obvious difference cannot be removed by employing the other group of the optical parameters in the calculations. This implies that the displacement model may not be adequate in accounting for the local distortion of Fe3þ in Al2O3:Fe3þ system. In order to simultaneously explain both the EPR second-order and fourth-order parameters D and ða 2 FÞ for Fe3þ in Al2O3:Fe3þ, in the following an elongation model will be proposed.

u2 ¼ u20 þ Du2 ;

ð15Þ

where

In fact, the real case is not as simple as the rigid limit, the oxygen in the oxygen triangle will be pushed by Fe3þ and Al3þ respectively along Fe– O and Al– O bond directions at the two sides of the oxygen triangle plane (See Figs. 1, 3). This will cause an appreciable expansion of the oxygen triangles and make the actual distortion angles Du1 ; Du2 less

4. Elongation model Since the transition-metal ion Fe3þ has an obviously larger ionic radius ðri ¼ 0:60 A) than that of ion Al3þ (rh ¼ 0:50 A) [11], it is reasonable to predict that when Fe3þ replaces Al3þ in Al2O3:Fe3þ, the Fe3þ ion will push the two oxygen triangles to move upwards and downwards, respectively, along threefold axis and this effect will induce the local sub-lattice, where the Fe3þ impurity ion is contained, to display a trend of elongation. We call this model as the elongation model. Here we use the approximate relationship [17]. 1 ðr 2 rh Þ; 2 i 1 R2 ¼ R20 þ ðri 2 rh Þ; 2 R1 ¼ R10 þ

ð14Þ

Fig. 3. Elongation Model of Fe3þ in Al2O3:Fe3þ system, Ri is the Fe–O bond length, and ui the angle between Fe–O bond and threefold axis, Li the distance between the O2- ligand and threefold axis.

W. Lu et al. / Journal of Physics and Chemistry of Solids 65 (2004) 1147–1151 Table 2 The ground-state splittings DE1 ; DE2 and the EPR zero-field splitting parameters a; D and ða 2 FÞ for Fe3þ ion in Al2O3:Fe3þ as a function of distortion angles Du1 ; Du2 in the elongation model and comparison with the experimental data, where 104 DE1 ; 104 DE2 ; 104 D; 104 ða 2 FÞ; are in units of cm21; 104 a ¼ 237 cm21 for Dq ¼ 1536 cm21 Du2

Du1

104 DE1

104 DE2

104 D

104 ða 2 FÞ

104 a

21.98

21.48 21.38 21.28 21.18 21.08 20.98

10171 10193 10215 10238 10260 10283

3930 3940 3950 3960 3970 3981

1672 1676 1679 1683 1686 1690

344 346 348 350 351 353

237 237 237 237 237 237

2 1.88

21.48 21.38 21.28 21.18 21.08 20.98

10389 10410 10432 10455 10478 10501

3997 4007 4017 4028 4038 4049

1709 1712 1716 1719 1723 1727

341 343 345 347 348 350

237 237 237 237 237 237

2 1.78

21.48 21.38 21.28 21.18 21.08 20.98

10605 10626 10648 10671 10693 10716

4064 4074 4085 4095 4105 4115

1745 1748 1752 1755 1759 1763

338 339 342 343 345 347

237 237 237 237 237 237

10451

4014

1719

339

236

Expt. [16]

than the predict values of rigid condition, i.e. the real distortion angles Du1 and Du2 are in the range: 2 1:68 , Du1 , 08; 2 2:88 , Du2 , 08:

ð16Þ

In order to accurately determine the distortion angles Du1 and Du2 of Fe3þ in Al2O3:Fe3þ, again the energy matrices [4] as well as the optical parameters B ¼ 660 cm21, C ¼ 3135 cm21, z ¼ 360 cm21 [5] and G4 ¼ 10140 cm21 are employed. The calculations of the EPR second-order and fourth-order parameters D and ða 2 FÞ as a function of distortion angles Du1 and Du2 are performed by diagonalizing the energy matrices. As shown in Table 2, for the distortion angles Du1 ¼ 21:18 ^ 0:18; Du2 ¼ 21:88; which are in accord with the predicted range of the rigid limit case, both the experimental values of the EPR parameters D and ða 2 FÞ can be satisfactorily explained. This means that the elongation model is a reasonable one for describing the local distortion of Fe3þ in Al2O3:Fe3þ. From our calculation, we have determined the local crystal structure parameters R1 ¼ 2:016 A, R2 ¼ 1:907 A, u1 ¼ 46:548; u2 ¼ 61:268 for Fe3þ in Al2O3:Fe3þ system.

1151

5. Conclusion The local lattice structure of Fe3þ in Al2O3:Fe3þ has been investigated by means of the theoretical analysis of EPR spectra of Al2O3:Fe3þ system. Both the second-order and fourth-order parameters D and ða 2 FÞ are used simultaneously in the structural study. Two distortion models, i.e. the displacement model and the elongation model, are respectively taken in analyses. It is demonstrated that the experimental EPR parameters D and ða 2 FÞ can be satisfactorily explained simultaneously only by the elongation model. This means that there exists an considerable elongation effect of the local lattice structure in Al2O3:Fe3þ. By diagonalizing the energy matrices the local structure parameters R1 ¼ 2:016 A, R2 ¼ 1:907 A, u1 ¼ 46:548 and u2 ¼ 61:268 for Fe3þ in Al2O3:Fe3þ system have been determined.

Acknowledgements The authors express their gratitude to Prof Gou QingQuan, Chengdu University of Science and Technology, for many helpful discussions. This project was supported by National Natural Science Foundation of China, No. 10374068.

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