Insight into Relationship between Crystal Structure and Crystal-Field

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Insight into Relationship between Crystal Structure and Crystal-Field Splitting of Ce Doped Garnet Compounds 3+

Zhen Song, Zhiguo Xia, and Quanlin Liu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b12826 • Publication Date (Web): 27 Jan 2018 Downloaded from http://pubs.acs.org on January 28, 2018

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The Journal of Physical Chemistry

Insight into Relationship between Crystal Structure and Crystal-Field Splitting of Ce3+ Doped Garnet Compounds Zhen Song1,2, Zhiguo Xia1, Quanlin Liu1,* 1

Beijing Key Laboratory for New Energy Materials and Technologies, School of

Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China 2

Departamento de Química, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Abstract

The common understanding of the negative relationship between bond lengths and crystal-field splittings is renewed by Ce3+ doped garnets in this work. We represent the contradictory relationship between structure data and spectroscopic crystal-field splitting in detail. A satisfactory explanation is given by expressing crystal-field splitting in terms of crystal-field parameters, on the basis of structural data. The results show that not only the bond length, but also the geometrical configuration has influence on the magnitude of crystal-field splitting. Also it is found that the ligand oxygen behaves differently with regards to multiple site substitution in garnet structure.

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1. Introduction Lanthanide aluminum/gallium garnet and its compositional derivatives with Ce3+ as the luminescence activator form a series of highly efficient luminescent materials. Besides the conventional application in cathode-ray display, fluorescent tube and scintillator, nowadays they are most widely used in the field of white light emitting diodes (WLED). Those phosphors are fabricated together with GaN chips, with the function of converting blue light (from GaN chips) to green-yellow light. The whole device could produce white light due to the combination of blue and green-yellow light. In order to improve the lighting quality, much effort has been paid to tune the spectroscopy. For example, with Ga replacing Al in Y3Al5O12:Ce3+, the emission changes gradually from yellow to green, which could be fabricated together with narrow-band red-emitting phosphors to realize wide gamut of liquid crystal display (LCD)1. Meanwhile, the red shift by substituting Y by Gd could reduce CCT (Correlated Color Temperature) and increase CRI (Color Rendering Index)2. In addition, the garnet phosphors are relatively cheap, chemically stable and easy for mass production. Until now, the garnet series phosphor is the dominant yellow phosphor used in WLED fabrication industry. The garnet series compound has the space group I a 3 d, and the crystal chemical formula can be expressed as (A)3{B}2[C]3O12. Generally, the larger lanthanides occupy the dodecahedral (A) site, while Al and Ga are accommodated in the octahedral {B} and tetrahedral [C] sites. In some cases, the occupation is rather complex, like the anti-site defect caused by Y, Lu entering {B} sites3-5 and the


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preferential occupation of Ga to [C] sites6-7. The importance of garnet phosphors arises from the spectroscopy of Ce3+ embedded in garnet lattice. The parity-allowed d-f transitions of Ce3+ are first recognized in 1960s in the pioneer work by Blasse et al.8-9, in which it is used as scintillator. Later, researchers began to pay attention to the absorption and emission spectra of Ce3+ in the garnet series, namely, Y3Al5O12(YAG), Y3Ga5O12(YGG), Gd3Al5O12(GdAG), Gd3Ga5O12(GdGG), Lu3Al5O12 (LuAG), Lu3Ga5O12(LuGG) and their solid solutions10-13. As will be shown in this work, the cell parameters, volumes of dodecahedron (A) and mean bond-lengths are consistent in trend with ionic radius change for different compositions. However, when the crystal-field splitting is taken into account, it gives self-contradictory relationships. For rare-earth elements doped phosphors involving d-f transitions, it is commonly recognized that the crystal-field splitting is closely related to the mean bond lengths between central lanthanides and ligands. In most cases this statement is true, as can be seen from the widely known d5 relation14-15 deduced from the octahedral ligands (Oh symmetry), which is

∆ = 10 Dq, D =

35Ze 2 2 4 〈r 〉 ,q = 5 4d 105

In this equation, ∆ stands for the crystal-splitting between eg and t2g levels of an d electron. Z is the ligand charge and d is the distance between central and ligand atoms. This relation serves as a bridge between crystal structure data of the host and luminescent spectroscopy of the doped lanthanides, such as Eu2+, Ce3+. It reveals a simple but intuitive rule: the shorter the bond length, the larger the crystal-field



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splitting. Researchers have applied this rule to explain the photoluminescent excitation(PLE)

peak

shifts

in

many

solid

solutions.

For

example,

in

M2MgSi2O7:Eu2+(M=Ca,Sr,Ba) series16, a red shift towards longer wavelength of PLE peaks is observed when the composition varies from Ba to Ca, with Ca-O bond lengths shorter than those of Ba-O. However, the simple relationship between bond length and crystal-field splitting is challenged by the lanthanide aluminum/gallium garnet, as mentioned above. For example, from YAG to YGG, the cell is expanding due to the larger Ga, and the crystal-field splitting is decreasing17, which is in accordance with expectation. On the other hand, the shrinkage of cell is observed from YAG to LuAG, while the crystal-field splitting is surprisingly reduced18-19. Some researchers name those compounds as reverse garnets14, 20. However, what is confusing is that we can observe the normal and reverse phenomena in the same compound, just as the example of YAG. This self-contradictory phenomenon greatly challenges the relationship between bond lengths and crystal-field splitting. The main interest of this work is focused on the explanation of normal/reverse garnet phenomena. One important attempt has been done by Seijo et al. using AIMPs (Ab Initio Model Potentials) methods21, which could yield all the five non-degenerate 5d levels. They find the results in good accordance with the normal/reverse phenomena. In addition, as pointed out by Dorenbos22, the type of polyhedron surrounding Ce3+ also influences the crystal-field splitting, and Wu et al. explore the correlation between the dodecahedral distortion and crystal-field splitting20. In this


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paper, we provide a more direct approach linking the garnet structure and crystal-field splitting by point-charge electrostatic model (PCEM) and simple perturbation calculation. The results will show that there exists no contradiction.

2. Analysis of garnet crystal structure Composition change of the above-mentioned garnets has no effect on the space group. Therefore, the structure change induced by dopant incorporation could be examined by checking lattice parameters and bond lengths. The detailed structural data is listed in Table 1. For the (A) site, there exist two kinds of bond lengths, therefore the mean bond lengths are taken into account. The relationship between composition, lattice parameter, bond length and volume of dodecahedra is displayed in Figure 1. It can be easily seen that the trends are in accordance with the ionic radius under 8 coordinated ligands, which is 0.977 Å for Lu3+, 1.019 Å for Y3+ and 0.938 Å for Gd3+23. The solid lines connecting the whole Al or Ga series show that the cell expands together with the bond length and dodecahedra as the sequence Lu < Y < Gd. Meanwhile, those garnets belonging to the same rare-earth element are connected by dashed lines, which also give increasing trend as the composition varies from Al to Ga. The above analysis show that it is the ionic size of different elements constituting garnets that determines the bond lengths between (A) site and ligands. In the next part we will check whether this statement holds for crystal-field splitting. Table 1. Crystal structure informaiton of garnet compounds Compounds Lattice Parameter (Å)

YAG 12.002

YGG 12.280

GdAG 12.114


GdGG 12.379

LuAG 11.906

LuGG 12.188


Bond Lengths (Å) Dodeca Mean Bond -hedron Lengths (Å) (A) Polyhedron Volume (Å3) ICSD #

Mean Bond Lengths (Å)Dodecahedral Volumes (Å3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2.439 2.306

2.428 2.338

2.522 2.353

2.527 2.354

2.383 2.276

2.393 2.303

2.372

2.383

2.437

2.440

2.330

2.348

22.895

23.052

24.895

25.026

21.621

22.027

4114524

2385225

19218426

19218126

2384625

2385025

25.0

GdAG

GdGG

24.5 24.0 23.5

YAG

23.0

YGG

22.5 22.0

LuAG

LuGG

21.5

GdAG

2.44

GdGG

2.42 2.40

YAG

2.38

YGG

2.36 2.34

LuAG

LuGG

2.32 11.8511.90 11.9512.0012.0512.10 12.1512.2012.25 12.3012.3512.4012.45

Lattice Parameter (Å) Figure 1. Mean bond lengths and dodecahedral volumes of garnet compounds as a function of lattice parameter

3. Energy level scheme of Ce3+ in garnet structures and crystal-field splitting For rare-earth elements doped phosphors, the dopant lanthanides generally occupy some specific sites. In garnet structure, Ce3+ is accommodated in the dodecahedral (A)


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site, with point symmetry D2. Within the 8 ligand-coordinated crystal-field potential, the 5-fold degenerate 5d levels will be totally split. The energy level scheme has been fully discussed27-29, and it is widely accepted that the degenerated energy levels are first split by Td crystal-field potential, with 2E (2-fold, low-lying) and 2T2 (3-fold, higher) levels, and then further split by D2 crystal-field potential. Researches are particularly interested in the two lowest 5d levels, because absorption transitions from the 4f ground levels to the two lowest 5d levels result in two PLE (photoluminescent excitation) peaks located in the visible-blue range. In this work, we treat the energy difference between those two peaks which is abbreviated as E12, as the representation of crystal-field splitting based on two reasons. First, the higher 5d levels may be immersed in the conduction band or located beyond the ultra-violet region9, which makes them difficult to detect. Second, as pointed by Dorenbos30, the location of PLE peaks are determined by both centroid shift and crystal-field splitting. A large survey of literature on Eu2+ and Ce3+ doped sulfide, oxide, fluoride and nitride compounds shows that the centroid shift is connected to composition, while the crystal-field splitting is mainly affected by geometrical configuration22, 31. Therefore, the energy difference, other than the energy location is discussed in the following sections. The E12 value could be easily determined by picking up the two peaks from PLE spectra in energy scale. The data is compiled in Table 2. Ogieglo et al. reported a crystal-field splitting larger than 8000 cm-1, which is far away from values of other references and is beyond consideration. The collection lacks data on YGG, with the possible reason that the luminescence of Ce3+ is quenched due to the down-shift of conduction band 32



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Table 2. Data of crystal-field splitting of Ce3+ doped garnets

Wavenumber (cm-1)

Crystal-field splitting

Ref

29239

21739

7500

Bachmann, et al.33

29239

21929

7310

Chen, et al. 2

8360

Ogieglo, et al. 34

YAG 29400

22000

7400

Blasse, et al. 9

YGG

28571

23810

4762

Ueda 35

GdAG

29762

21253

8509

Chen, et al. 2

28735

23364

5371

Ogieglo, et al. 36

28700

23500

5200

Kaminska, et al. 37

28233

21138

7100

Ogieglo, et al. 34

GdGG LuAG


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8600

5400

8400

GdAG

Ogieglo, et al. Bachmann, et al. Chen, et al. Chen, et al. Blasse, et al. Ogieglo, et al. Ogieglo, et al. Ueda Kaminska, et al.

5300

8200 5200 8000 5100

7800

YAG

7400

4900

7200 7000 2.32

GdGG

5000

7600

Crystal-field splitting, ∆ (cm-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4800

2.33

7500

YGG

(a)

LuAG 2.34

2.35

2.36

2.37

2.38

2.39

(b) 4700 2.380

2.40

2.385

2.390

2.395

2.400

2.405

2.410

2.415

2.420

9000

YAG

GdAG

(c)

(d)

8500 7000 8000 6500

7500

6000

7000 6500

5500 6000

GdGG

5000

5500

YGG 4500 2.370

2.375

2.380

2.385

5000 2.395

2.400

2.405

2.410

2.415

Bond lengths (Å)

Figure 2. Relationship between crystal-field splitting and bond lengths compiled from references

Investigation into the relationship between bond lengths and crystal-field splitting of Ce3+ doped garnets, as shown in Figure 2, produces self-contradictory results. For the aluminum-garnet series, E12 is increasing together with the cell expansion in the sequence of Lu < Y < Gd, as shown in Figure 2 (a). This reverse-garnet phenomenon is also observed in Figure 2 (b) for YGG and GdGG. However, the normal-garnet phenomenon also exists, as the cases of Figure 2 (c) and (d), in which E12 decreases with larger Ga substituting Al. This contradiction puts up great challenge to the simple relationship between bond length and crystal-field splitting, and we will show in the next section that the origin arises from the limitation of d5 relation.



4. Crystal-field analysis of Ce3+ in garnet structure Crystal-field theory has been widely applied to rare-earth doped crystals38-39. By fitting crystal-field parameters to spectroscopic measurements, the energy levels of doped rare-earth elements could be deduced40. The simple d5 relation is deduced under Oh symmetry, i.e. center of 8-coordinated octahedron. The symmetry elements in Oh point group will eliminate all the 2nd rank crystal-field parameters in the crystal-field potential41-42. As a result, for d electron the Oh crystal-field potential could be expressed using only one 4th rank crystal-field parameter, i.e. B40. If expressed by the point-charge electrostatic model (PCEM), B40 has the form of d5 in the denominator. However, when the site symmetry degrades, the symmetry limitations on the 2nd crystal-field parameters are broken, and they become non-zero, which will introduce d3 together with d5. As mentioned above, for garnet structure, the dopant Ce3+ is accommodated in the 8-coordinated (A) site, with point symmetry D2. The crystal-field potential is expressed as41 V (D2 ) =  B20U 02 + B22 (U 22 + U −22 )  l C 2 l +  B40U 04 + B42 (U 24 + U −42 ) + B44 (U 44 + U −44 )  l C 4 l

Therefore, it is expected that the 2nd rank crystal-field parameters will be included in the analytical expression of split energy levels. In this case, the simple d5 relationship takes no more effect. We will follow a reported routine17 to treat the crystal-field splitting of Ce3+ doped garnets. In brief, the perturbation matrix is constructed according to the D2 crystal-field potential, and then the eigenvalues are obtained by matrix


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diagonalization. The important point lies in how to treat crystal-field potential and spin-orbit coupling. Unlike transition-metal elements, for rare-earth elements the spin-orbit coupling is significant. If spin-orbit coupling is treated prior to crystal-field potential, the energy levels of 5d1 electron are first split into 2D5/2 and 2D3/2 terms, which will be further split under the crystal-field potential. However, this treatment is inconsistent with the real case, because the energy difference between the two 2D terms is 5ξ/2 (ξ , the spin-orbit coupling constant, has a value of 991 cm-1 for 5d electron of Ce3+)43, much smaller than the energy level separation. If the energy levels are first split by crystal-field potential and after that spin-orbit coupling is considered, the final split energy will be only related to ξ, which means the energy level separation keeps unchanged among different compounds. This is also inconsistent with the real case. In fact, in the correct method the spin-orbit coupling and crystal-field potential should be considered simultaneously, but this treatment will lead to solving a 5-power secular equation during the diagonalization process, which seldom gives analytical results. A more realistic routine is to assume cubic field >> the spin-orbit interaction>>D2 crystal-field28, and by this method we have shown the decrease of crystal-field splitting from YAG to YGG, as explained in Appendix of Ref 17. In this work, the other garnets are treated in the same way, and the results are listed in Table 3. Interpretation of those results provide convincing explanation for the above mentioned self-contradictory phenomena. For the aluminum and gallium series, it is clear that E12 increases as Lu < Y < Gd. For LnAG/LnGG paris, i.e. LuAG/LuGG, YAG/YGG and GdAG/GdGG, E12 drops as Ga substitutes Al. Those



results are in accordance with Figure 2 and the contradiction disappears. It should be noted that the sequence is related to the values of and , for which a coarse estimation is provided in the Supporting information.


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Table 3. Expressons of crystal-field parameters and crystal-field splittings of garnet compounds LuAG

YAG

GdAG

LuGG

YGG

GdGG

B20

−0.01100e2 〈r 2 〉 Z

−0.01294e2 〈r 2 〉 Z

−0.01824e2 〈r 2 〉 Z

−0.004151e2 〈r 2 〉 Z

−0.005184e2 〈 r 2 〉 Z

−0.01253e2 〈 r 2 〉 Z

B22

0.01347e2 〈r 2 〉 Z

0.01585e2 〈r 2 〉 Z

0.02234e2 〈r 2 〉 Z

0.005085e2 〈r 2 〉 Z

0.006349e2 〈 r 2 〉 Z

0.01534e2 〈r 2 〉 Z

B40

0.02201e2 〈r 4 〉 Z

0.02000e2 〈r 4 〉 Z

0.01728e2 〈r 4 〉 Z

0.02140e2 〈r 4 〉 Z

0.02005e2 〈r 4 〉 Z

0.01694e2 〈r 4 〉 Z

B42

0.01228e2 〈r 4 〉 Z

0.01106e2 〈r 4 〉 Z

0.00872e2 〈r 4 〉 Z

0.01312e2 〈r 4 〉 Z

0.01172e2 〈r 4 〉 Z

0.01040e2 〈r 4 〉 Z

B44

0.003865e2 〈r 4 〉 Z

0.003600e2 〈 r 4 〉 Z

0.003737e2 〈r 4 〉 Z

0.002870e2 〈 r 4 〉 Z

0.003125e2 〈r 4 〉 Z

0.002259e2 〈 r 4 〉 Z

57584.7 B202

57618.0 B202

57560.00B202

2 57600.00 B20

57600.0 B202

57533.9 B202

+34232.5B20 B40

+33926.3B20 B40

+31065.01B20 B40

+37499.39 B20 B40

+35797.0 B20 B40

+37525.9 B20 B40

+5089.22 B

+4995.75B

+4190.97B

+6104.26 B

+5563.02 B

+6119.87 B402

2 40

(E12)2

 239.97 B20  ≈   +71.34B40 

2 40

2

  2.64〈 r 2 〉   =  Ze 2   4   −1.57〈 r 〉   

 240.04 B20  ≈   +70.68 B40  2

2 40

2

 2  3.11〈 r 2 〉   =  Ze   4   −1.41〈 r 〉   

 240.00 B20  ≈   +64.73B40  2

2 40

2

 2  4.38〈 r 2 〉   =  Ze   4   −1.20〈 r 〉   

 240.00 B20  ≈   +78.13B40  2

2 40

2

 2  0.996〈 r 2 〉   =  Ze   4   −1.67〈 r 〉   


 240.00 B20  ≈   +74.59 B40  2

2

 2  1.24〈 r 2 〉   =  Ze   4   −1.50〈 r 〉   

 239.86 B20  ≈   +78.23B40  2

2

 2  3.01〈 r 2 〉   =  Ze   4   −1.33〈 r 〉   

2


A close examination of Table 3 shows that B20 changes magnitude sharply among different compounds, while the change of B40 is relatively small. For both Al and Ga garnet series, the magnitude of B20 increases as the sequence Lu < Y < Gd. The same trend also holds for every Al/Ga pair, just in accordance with E12. Meanwhile, when Ga completely substitutes Al, the magnitude of B20 drops nearly half. To get a more vivid understanding, the contribution of every ligand to B20 of all the compounds is summarized in the Supporting Information. It is easily seen that for every compound only ligand 4 and 5 have positive contribution. These two ligands are related by one 2-fold axis. They share the same bond lengths and have the same contribution to B20. In addition, ligand 4 and 5 are more close to the polar axis (z axis) than the other ligands. Based on these facts, we consider the sum of contributions of ligand 4 and 5 as a whole, whereas the sum of contributions from the other ligands is considered as the counterpart for comparison. They are listed in Table 4, and plotted in Figure 3. For both positive and negative contributions, the magnitudes are in accordance with the bond lengths. Figure 3 (a) and (b) shows that for both Al and Ga series, Lu garnet has the largest magnitudes of the two items, and Gd garnet has the smallest. This is reasonable because B20 is reverse proportional to r3, which could be seen from the PCEM expression e 2 〈 r 2 〉 Z (1 + 3cos 2θ ) B20 = 4r 3

However, the difference between the positive and negative parts has reverse trend in magnitude. For Gd garnet, although the magnitudes of positive and negative contributions are the smallest,


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the difference is the largest, as shown by the blue triangle line in Figure 3 (a) and (b). As a result, Gd garnet has the largest B20 among Al or Ga series. For the LnAG/LnGG pairs, Figure 3 (c) shows that the magnitude of the negative contribution also decreases as cell expands. However, the positive contribution has a rising trend for all the Al/Ga pairs, despite the fact that bond lengths are stretching from Al to Ga. This arises from the effect of polar angle (θ) which appears in the nominator of B20 expression. When θ < 45 °, B20 is positive and has a larger magnitude at smaller polar angles. From Figure 3 (d), it is known that all the Ga garnet series have smaller polar angles than Al series. In the case of LnAG/LnGG pairs, the polar angle dominates the magnitude of positive contribution to B20. As a result, with smaller negative contribution and larger positive contribtuion, the magnitude of B20 shrinks from Al to Ga, which leads to the decrease of E12. It should be noted that the results of crystal-field contributions of every ligand depend on the axis choice. Under different coordinate-systems the values may change, but the overall result remains the same due to physical equality. Table 4. Polar angle (θ), positive and negative B20 contributions of Al and Ga series garnet Garnet Compound Polar angle of ligand 5 (°) Sum of 4 and 5 (Positive Contribution) 2 2 B20/ e 〈 r 〉 Z Sum of the other (Negative Contribution) Total

LuAG 11.29

YAG 11.24

GdAG 11.59

LuGG 10.54

YGG 10.79

GdGG 10.24

0.139

0.131

0.117

0.139

0.132

0.118

-0.150

-0.144

-0.135

-0.143

-0.138

-0.130

-0.011

-0.013

-0.018

-0.0041 -0.0052 -0.0125



0.16

0.00

0.16

0.14

-0.02

0.14

0.00 -0.02

0.12

-0.04

0.12

-0.04

0.10

-0.06

0.10

-0.06

0.08

-0.08

0.08

-0.08

0.06

-0.10

0.06

-0.10

0.04

-0.12

0.04

-0.12

0.02

-0.14

0.02

-0.14

-0.16

0.00

0.00

LuAG

YAG

(b)

B20 coefficient

B20 coefficient

(a)

-0.16

GdAG

LuGG

0.16

YGG

GdGG

0.00

(c)

(d)

11.6

0.14

-0.02

0.12

-0.04

0.10

-0.06

0.08

-0.08

0.06

-0.10

0.04

-0.12

0.02

-0.14

0.00

-0.16

Polar angle degree (°)

11.4

B20 coefficient

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11.2 11.0 10.8 10.6 10.4 10.2

LuAG

LuGG

YAG

YGG

GdAG Sum of 4+5

GdGG Sum of the other

LuAG Total

LuGG

YAG

YGG

GdAG

GdGG

Polar angle of ligand 5

Figure 3. Absolute value of B20 coefficient for: sum of ligand 4 and 5, blue triangle; sum of the other ligands, red square; total sum of all the ligands, red circle; polar angle of ligand 5, pink diamond. (a) and (b) represents the Al and Ga garnet series, respectively, while (c) and (d) represents LnAG/LnGG paris.

5. Structure-property correlations in garnet structure The eight coordinated ligands of the dodecahedral (A) site could be categorized into two groups, as colored in red and blue in Figure 4 (a). Although they are crystallographically equivalent, ligand 1, 2, 6, 7 share the same bond length, while ligand 3, 4, 5, 8 share the other bond length. The dodecahedron could be thought as a distorted cube, as shown in Figure 4 (b),


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with ligand 1-4-7-8 constituting the bottom face, and ligand 2-3-6-5 the top face. Under the point symmetry D2, some of the edges share the same distances, which are d23 = d56 = d18 = d47, d14 = d78 = d25 = d36, d21 = d67 and d58 = d34. In addition, the equivalent edges have the same connectivity with polyhedrons, as illustrated in Figure 4 (c). In the top and bottom face, the four edges are shared by octahedron and another dodecahedron successively. For the four edges connecting the top and bottom faces, edge 2-1 and 6-7 are shared by tetrahedrons, while edge 5-8 and 3-4 have no share with other polyhedrons. In Figure 4 (d), the distortion could be visualized more clearly from the top view of the cube. In addition, distortion of the cube has been adopted by Euler and Bruce to investigate the crystal chemistry of garnets25. Wu et al. discovered the ratio between d88 and d18 could be used to characterize the crystal-field splitting of Ce3+ in garnets20, while d88 and d18 represent the O-O distances shared by adjacent dodecahedron and tetrahedron, respectively. In this work, d88 and d18 are equivalent to the distances between ligand 2, 3 and 2, 1 as shown in Figure 4 (c). Consequently, in Figure 4 (d) the distortion of the cube could be represented by d23/d21, which are plotted for both Al and Ga garnet series in Figure 5. For Al/Ga series, incorporation of larger cations into dodecahedral (A) site increases the ratio. On the contrary, for LnAG/LnGG pairs larger Ga substituting Al results to a decrease in d23/d21. It is evident that the ratio between d23 and d21 has the same trend with crystal-field splitting (Figure 2) and crystal-field parameter (Figure 3). However, to the best of our knowledge it is difficult to connect crystal-field splitting with cube distortion. As will be shown in this section, the similarity in the varying trend of crystal-field splitting and cube distortion arises from the movements of dodecahedral ligands through garnet compounds.



Figure 4. (a) Coordinated ligands of dodecahedral (A) site. The crystallographically equivalent 8 oxygen atoms could be grouped into two categories according to the bond lenths. They are rendered by different colors; (b) Perspective view of the distorted cube of the dedecahedral ligands; (c) Schematic of the adjacent dodecahedron, octahedron and tetrahedron with respect to the oxygen ligands; (d) Top view of the distorted cube of the dodecahedral ligands. Solid lines indicate shared edges with another dodecahedron, dashed lines octahedron and dotted lines tetrahedron; (e) Ligand movement pattern for LuAG to YAG; (f) Ligand movement pattern for LuAG to LuGG. The arrows indicate the moving direction of ligands when the garnet compound change. The magnitude of the arrow has been magnified 30 times for a better visualization. Plotted by VESTA44

It is known from the above section that both incorporation of larger size lanthanides and substituting Al for Ga have the effect of expanding the cell and dodecahedron volume. However, there exist subtle differences between how the oxygen ligands respond to different site


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substitutions. A detailed study of the dodecahedral ligand coordinates of garnet compounds reveals that the ligand movement has two distinct patterns with regards to site substitutions. In Figure 4 (e) and (f), the evolvements from LuAG to YAG and LuAG to LuGG are selected to elucidate the two ligand movement patterns. The arrows indicate the ligand movement between different garnet compounds. For the (A) site substitution, the ligand movement shown in Figure 4 (e) indicates a much larger elongation of distance between O2 and O3 than that between O1 and O2. This pattern is in accordance with the dodecahedron inflation from LuAG to YAG, considering the fact that edge 2-1 is shared by tetrahedron, while edge 2-3 is shared by another dodecahedron. Therefore, d23/d21 is increasing through Lu-Y-Gd among Al and Ga garnet series. However, for the {B} and [C] sites substitutions, which are referred to Ga substitution for Al from LuAG to LuGG in Figure 4 (e), they share a quite different mode. The ligand movement pattern indicates a further separation between O1 and O2, but an approach between O2 and O3. This means inflation of octahedron and tetrahedron simultaneously squeezes dodecahedron. As a result, we expect a shorter d23 and longer d12, then finally a smaller d23/d21. Meanwhile, it is noted that ligand 4 and 5 approach closer to z axis in the second movement pattern, which accounts for the smaller polar angles of Ga series garnet in Figure 3 (d).



1.12

1.12

1.10 1.08 1.06

d23/d21

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1.05

1.05 1.04 1.02

1.02

1.00

0.991

0.98

0.971 0.96 LuAG

LuGG

YAG

YGG

GdAG

GdGG

Figure 5. Cube distortion (d23/d21) for garnet compounds. Red squares belong to the Al series, while blue circels belong to Ga series. Data is extracted by VESTA.

From this point of view, it is known that the ligand oxygen atoms respond in different ways regarding to different site substitutions. Although the whole effect is the same, i.e. larger cell and longer bond lengths, the two ligand movement pattern leads to the reverse trend of crystal-field splitting and cube distortions. The normal and reverse phenomena on crystal-field splitting phenomena of garnet structures could be understood from the above treatment. However, it should be noted that PCEM is a coarse model, because the ligands are regarded as negative point-charges in this model, which


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deviates from the real case. Moreover, the bond lengths are extracted from pure garnet structures without dopant Ce3+, and it is known that incorporation of larger Ce3+ will cause cell expansion, which means the real bond lengths should be larger than those used in this work. Nevertheless, PCEM could be regarded as the first approximation, and it could be applied to explain the crystal-field splitting in Ce3+ doped garnets.

6. Conclusions This work is devoted to explain the crystal-field splitting in Ce3+ doped garnet compounds. Generally, incorporation of larger size atoms will expand the cell and prolong the bond lengths. This includes the successive substitution of Lu, Y, Gd for dodecahedral (A) sites, and Ga substituting Al. The crystal-field splittings of garnet compounds show reverse trends. For Al or Ga garnet series, E12 increases in the sequence LuAG

Insight into Relationship between Crystal Structure and Crystal-Field

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