Elastic properties of YAG: First-principles calculation and experimental investigation

Solid State Sciences 14 (2012) 1327e1332

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Elastic properties of YAG: First-principles calculation and experimental investigation Zuocai Huang a, *, Jing Feng a, b, Wei Pan a a

State Key Laboratory of New Ceramics and Fine Processing, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Key Laboratory of Advanced Materials of Precious-Nonferrous Metals, Education Ministry of China and Key Laboratory of Advanced Materials of Yunnan Province, Kunming University of Science and Technology, Kunming 650093, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 June 2011 Received in revised form 28 April 2012 Accepted 9 July 2012 Available online 20 July 2012

The hardness and elastic stiffness of Y3Al5O12 (YAG) were investigated by first-principles calculations and experiments. The mechanical properties including the second-order elastic coefficients, hardness, bulk modulus, Young’s modulus and shear modulus were calculated by density functional theory (DFT). The calculated results were in good agreement with the experimental values. The hardness of YAG is mainly attributed to AlteteO bonds. The elastic anisotropy of YAG was discussed. Zener anisotropy parameter of YAG is close to unit and its universal anisotropy index is very close to zero, which indicates the structure of YAG is nearly centrosymmetric. The longitudinal and transverse sound velocities and Debye temperature were also investigated. Ó 2012 Elsevier Masson SAS. All rights reserved.

Keywords: Y3Al5O12 Density functional theory (DFT) Hardness Elastic anisotropy

1. Introduction Yttrium aluminum garnet (Y3Al5O12) is the most important solid-state laser host material, which has been widely used in the areas of medicine and industry since its discovery in 1964 [1]. Apart from the ideal spectroscopic properties of the rare-earth ions in YAG crystal, its good mechanical properties (hardness and general stability against chemical and mechanical changes), good thermodynamic properties (low-thermal expansion and high thermal conductivity), and good optical properties (high-optical transparency, low-acoustic loss, high threshold for optical damage) all contribute to its success as the most widely used laser material [2]. YAG is also one of the most creep-resistant oxides and has important applications in high-temperature ceramic composites [3,4]. Owing to its numerous practical applications, a thorough understanding of the mechanical properties is very important. To improve existent applications and develop new instruments, knowledge of the mechanical properties, for instance elastic constants or hardness, is a necessity [5]. Several studies with the goal of understanding elastic stiffness of YAG have been reported [6e9]. But the elastic anisotropy of YAG was rarely discussed. It is known that the elastic response of a single crystal is almost

* Corresponding author. Tel.: þ86 10 62782283; fax: þ86 10 62771160. E-mail address: [email protected] (Z. Huang). 1293-2558/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.solidstatesciences.2012.07.008

anisotropic. Ledbetter and Migliori [10] highlighted the importance of the elastic anisotropy in practical applications such as phase transformations and dislocation dynamics. For its complicated structure, few researches on Y3Al5O12 were conducted by DFT. Xu and Ching [11] computed the electronic structure, the lattice constant and the bulk modulus. Chen et al. [12] calculated the local atomic structures, electronic structure and formation energies of vacancies by ab initio calculation. Marquardt et al. [6] used ab initio calculations to calculate the elastic properties of Y3Al5O12. They adopted the experimental relative atomic positions within the unit cell during calculations, but didn’t attempt the calculation of the elastic stiffness at the atomistic level. To predict the structure of YAG from the atomistic level by first principles is very important for further calculations. In this work, an effort was made to understand the mechanical properties, especially the elastic anisotropy of YAG at the atomistic level by first principles. 2. Materials and methods Yttrium aluminum garnet investigated in this work was grown by the Czochralski method. The pure Y2O3 and Al2O3 powders were appropriately dried and weighed according to the formula Y3Al5O12. After the compounds were ground and mixed, they were pressed into bulks and put into an alumina crucible. The bulks were sintered at 1200
C for 10 h in air, and then were loaded into an

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Z. Huang et al. / Solid State Sciences 14 (2012) 1327e1332

iridium crucible for crystal growth. High-purity nitrogen gas was used as a protective atmosphere. To prevent the crystal from cracking, it was cooled slowly from 1200
C to room temperature. Plates parallel to the (100) plane (shown in Fig. 1) were cut from the crystals, ground down to a thickness of 1 mm, and then polished for ultrasonic and mechanical experiments. Density-functional calculations were applied to get the theoretical crystal parameters of Y3Al5O12, which is implemented through the Cambridge Serial Total Energy Package (CASTEP) Program [13]. The interactions between valence electrons and the ionic core were represented by ultra-soft pseudo-potentials. The exchange-correlation energy was calculated by GGA in terms of PBE schemes [14]. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme [15] was used in geometry optimization. The valence electron configurations considered in this study included O2s22p4, Al3s23p1 and Y4s24p64d15s2. We used a plane wave basis with a cut-off of 600 eV to expand the pseudo valence wave function. The special points sampling integration over the Brillouin zone was realized using the Monkhorst-Pack method with a 6  6  6 special k-points mesh [16], where self-consistent convergence of the total energy is at 105 eV/atom and the maximum force on the atom is within 0.01 eV/Å. The elastic constants were determined from first-principles calculations by strain-stress method, that is applying a set of given homogeneous deformations with a finite value and calculating the resulting stress with respect to optimizing the internal atomic freedoms, as implemented by Milman and Warren [17]. In order to get the intrinsic hardness by qualitatively evaluating the effect of orbital electrons on the chemical bonding of the compounds, we further calculated the population characteristic using Mulliken’s method [18].

3. Results and discussions 3.1. Crystal structure and electronic structure Yttrium aluminum garnet crystal has a chemical formula A3B0 2B00 3O12, where A, B0 , B00 are metal ions occupying different symmetry sites, which is shown in Fig. 2. It has a bcc structure (space group Ia3d) with 160 atoms in the cubic cell. The Y ions occupy the 24 (c) sites and each is dodecahedroally coordinated to eight O. The O atoms occupy the 96 (h) sites whose locations

Fig. 2. The crystal structure of YAG (Y3Al5O12).

depend on three structural parameters x, y, and z and are different for different garnet oxides. There are two different sites for Al, Aloct (B0 ) atoms occupy the 16 (a) site with an octahedral point symmetry (C3i) and Altet (B00 ) atoms occupy the 24 (d) sites with a tetrahedral point symmetry (S4) [19]. To investigate the ground-state properties, the equilibrium lattice configuration of Y3Al5O12 was computed at first. Optimized lattice constants and the x parameter are listed in Table 1, as well as experimental results [20] and other first-principles calculations [21]. The computed lattice constant is 12.286 Å, which is 2.4% higher than the previously reported data. This deviation can stem from the use of GGA approximation. The density of states (DOS) and projected density of states (PDOS) of YAG were plotted in Fig. 3. We can find the DOS can be mainly divided into three parts. The deeper than valence band extending from 23 eV to 15 eV is of the combination of O s and Y p states. The valence band from 6.5 eV to 0 eV is mainly the contribution of O p states. The conduction band extending from 4.8 eV is mainly composed of Y s, Y p, Al s and Al p states. 3.2. Mulliken populations and intrinsic hardness

(800)

Intensity(a.u.)

(400)

YAG

In Table 2, the bondlength and orbital populations are shown. It is shown both YeO and AleO bonds show strong covalent characters because of the positive overlap populations. Y ion has less charge transferred to O than Al, so AleO bonds have more covalent characters than YeO bonds. In order to investigate the bonding characteristics, we focused on the valence electron density of YAG on the (011) plane as shown in Fig. 4. It shows that the distribution of electronic density between Al and O atoms is higher than that between Y and O atoms, which indicates that the bond AleO has more covalent components than YeO and keeps in accordance with Mulliken population analysis. The bond order (overlap population)

Table 1 Lattice parameter of Y3Al5O12.

20

40

60

2θ (deg) Fig. 1. The X-ray diffusion pattern of YAG plates.

80

Source

a (Å)

x (O)

y (O)

z (O)

Exp. [20] LDA [21] PBE [21] This work

12.000 11.691 12.114 12.286

0.0306 0.0290 0.0306 0.0315

0.0512 0.0510 0.0519 0.0488

0.1500 0.1499 0.1491 0.1490

Z. Huang et al. / Solid State Sciences 14 (2012) 1327e1332

1329

Fig. 4. Valence electron density distribution on the (011) plane of Y3Al5O12.

h m   m i1=P nm m n Hn ¼ P Hn Fig. 3. Calculated density of states of YAG (a) and projected densities of states of the atoms of the unit cell O (b), Al (c), Y (d).

is a simple qualitative measure of the strength of the bond between a pair of atoms. Compared to all other pairs in YAG crystals, the bond order for AlteteO is relatively large. What is more, the Al tetrahedral sites take up 60% of the Al sites, which contribute to the good mechanical properties of YAG crystals. Table 2 also shows the intrinsic hardness of YAG crystals. From the results of Mulliken population analysis of crystals, the intrinsic hardness of YAG can be quantitatively estimated based on Gao’s work [22]. For complex multibond compounds, the hardness of the m-type bond can be calculated as follows:


5=3 m m Hy ðGPaÞ ¼ AP m yb ðdm Þ3 Vcell

ymb ¼ Ph v

ðdv Þ3 Nby

(1)

i

(2)

Where the value of the proportional coefficient A is suggested as m 740, P m is the Mulliken overlap population of the m-type bond, yb is m the volume of a bond of type m, d is the bondlength of type m, Nby is the bond number of type n, Vcell is the cell volume of each Y3Al5O12 compound. The intrinsic hardness for each chemical bond is calculated and the hardness of the crystal is obtained by an average of the hardness of all binary systems in crystals:

(3)

Here nm is the number of m-type bonds composing the actual complex crystal. For Y3Al5O12, the hardness is calculated by:

i1=196 h 48 48 48 48 HðY3 Al5 O12 Þ ¼ HAl H H H Y Y eO eO Al eO eO 1 2 tet oct

(4)

The measured Vickers hardness of YAG is 18 GPa [30] and the calculated hardness is 18.2 GPa. So the calculated hardness is in good agreement with experimental results. From Table 2, it is clear that chemical bonds with larger band order and shorter bonder length is more favorable for higher hardness. 3.3. Elastic properties The calculated second-order elastic constants of YAG are summarized in Table 2. For a cubic crystal, there are three independent elastic constants: C11, C12 and C44. C11 represents the uniaxial deformation along [001] direction and C12 is a mixture of s11 and s22 (or the pure shear stress at (110) plane along [11 0] direction). In general, the calculated C11 is the largest elastic constant among the three independent parameters. As shown in the table, the calculated elastic constants are a little smaller than others, but the discrepancy is within 10 GPa. That means our calculated results keep in good agreement with experiments and other calculations. Bulk modulus B is shown in Table 2, which measures the resistance of a material to volume change and provides an estimate of its elastic response to hydrostatic pressures

Table 2 Mulliken population analysis and hardness of Y3Al5O12. Atom

s

p

d

Total

Charge (e)

Bond

Population

Length (Å)

Nu

U (Å3)

Vbu

Hu

H Cal.

Exp.

O Aloct Altet Y

1.84 0.51 0.50 2.25

5.16 0.95 0.90 6.17

0 0 0 1.20

7.00 1.46 1.40 9.62

1.00 1.54 1.60 1.38

AlteteO AlocteO Y1eO Y2eO

0.49 0.34 0.19 0.21

1.7465 1.9133 2.3033 2.4534

48 48 48 48

936.73

5.15 3.36 6.26 7.30

84.9 37.3 8.3 6.7

18.2

18 [30]

H

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Z. Huang et al. / Solid State Sciences 14 (2012) 1327e1332

Table 3 Elastic coefficients (GPa), bulk modulus (GPa), shear modulus (GPa), Yong’s modulus (GPa), Poisson’s ratio (s), anisotropic properties (AZ and AE). Source

Method

C11

C12

C44

B

E

G

s

B/G

AZ

AE

AU

Bush et al. [27] Spencer et al. [8] Alton et al. [7] Yogurtcu et al. [28] Chistyi et al. [29] This work Experiment

Pair potential Ultrasonic Ultrasonic Ultrasonic Brillouin DFT(GGA) Ultrasonic

330.3 333.2 334 328 333 309.5 327.6

164.7 110.7 111 106 114 106.8 112.2

131.2 115 115 113 114 103.7 109.0

220 185 185 180 187 174.4a, 182.6b 180.6

220.7 277.9 278.6 275.9 274.8 254.7 267.4

109 113 114 113 112 100.7 107.0

0.333 0.249 0.25 0.24 0.26 0.256 0.255

2.66 1.66 1.66 1.62 1.71 1.73 1.677

1.58 1.03 1.03 1.03 1.04 1.02 1.01

0.49 0.03 0.03 0.03 0.04 0.02 0.01

0.26 1.3E-3 1.3E-3 7.7E-4 1.9E-3 6.3E-4 1.7E-4

a b

From B0 ¼ (C11 þ 2C12)/3. From Birch equation of state.

[23]. In order to get comparison, we also calculated the EOS of YAG and get the bulk modulus from the EeV curve fitted by Birch EOS. The bulk modulus values from elastic constants, Birch EOS and experiment keep in accord with one another. We also compare the calculated Young’s modulus with the experimental values and other calculations. The calculated results in this paper are all smaller than others. The reason for this discrepancy is that the GGA-based DFT (density functional theory) calculation typically underestimates the interatomic bonding and deduces lower elastic constants and mechanical parameters, especially for Young’s modulus E (E increases with higher binding energies and shorter bondlength). In the cubic crystal, if the interatomic force is central, then C12] C14. In Table 2, it is shown that the C12 of YAG is very close to C44, but not equal. In some results, the former one is a little smaller than the latter one, while the other results show the opposite direction. If the Cauchy relation (C12]C14) exists in cubic crystals, the Poisson ratio is rigidly equal to 0.25. From Table 2, we can find the Poisson ratio of most results is close to 0.25. There are two types of anisotropic index to characterize the elastic anisotropy of cubic crystal; one is the widely used Zener’s anisotropic index [24] and the other is the universal anisotropic index proposed by Ranganathan et al. [25]. Zener anisotropic ratio can be only applied to cubic crystal classes, and the universal elastic anisotropic index is applicable for any type of crystal structure. Zener anisotropic ratio is calculated by Eqs. (5) and (6)

AZ ¼ 2C44 =ðC11  C12 Þ

(5)

AE ¼ ðC11  C12  2C44 Þ=ðC11  C44 Þ

(6)

Z

In the above equations, G and B are the shear modulus and bulk modulus; V and R represent the Voigt and Reuss estimations for B and G, respectively. AU is identically zero for locally isotropic single crystals. The departure of AU from zero defines the extent of single crystal anisotropy, while the deviation of AZ from unity refers to the extent of anisotropy. From the calculated Zener anisotropic ratios and the universal elastic anisotropic index in Table 3, the proportion of YAG’s anisotropic nature is low. As was already discussed above, for YAG, C44 sC12 and the corresponding interatomic forces are not centrosymmetric. The anisotropic property of shear modulus (G) for cubic crystal can be measured by the difference between C44 and ð1=2ÞðC11  C12 Þ, where C44 and ð1=2ÞðC11  C12 Þ represent the shear modulus along the [100] and [110] direction, respectively. We defined G100 ¼ C44, G110 ¼ ð1=2ÞðC11  C12 Þ and G0 ¼ 1/2(G100 þ G110) for convenience. For an isotropic crystal, G100 ¼ G110. We plotted the results using polar coordinates for YAG as shown in Fig. 5. The x and y directions in the figure are along the [100] and [110] crystal orientations respectively. The angle q refers to the direction difference between the diagonal line and G0 . The plot in Fig. 5 shows that the shear modulus of YAG is anisotropic. For the angle q is very small, we can think of YAG has high centrosymmetry in comparison with other single crystals. We also analyzed the Young’s modulus along different directions and it is shown is Fig. 6. From Fig. 6, we can see the 3D curve of Young’s modulus is nearly a sphere and the projection of Young’s modulus on (001) plane is nearly a circle, so it also means YAG is highly centrosymmetric.

E

If A ¼ 1 (A ¼ 0), the material is isotropic, which means that the modulus is independent on orientation. For cubic lattice, Zener anisotropy ratio also determines the extreme of Young’s modulus: for example, for AZ < 1, the maximum Young’s modulus would be along the [100] direction; while for AZ > 1, maximum Young’s modulus would be in the [111] direction. The universal elastic anisotropic index is given by AU [25].

AU ¼ 5

GV BV þ 60 GR BR

(7)

ðC11 þ 2C12 Þ 3

(8)

BV ¼ BR ¼

GV ¼

C11  C12 þ 3C44 5

GR ¼

5ðC11  C12 ÞC44 4C44 þ 3ðC11  C12 Þ

(9)

(10) Fig. 5. Anisotropic shear modulus in different crystal directions of Y3Al5O12.

Z. Huang et al. / Solid State Sciences 14 (2012) 1327e1332

1331

Fig. 6. (a) Directional dependence of Young’s modulus and (b) Projection of Young’s modulus on (001) plane for Y3Al5O12. The units are in GPa.

3.4. Debye temperature

4. Conclusions

We measured and calculated sound velocities of both longitudinal and transverse waves along [100] direction. The equations used for calculations are shown as the following:

In summary, the crystal structure, hardness, elastic stiffness and Debye temperature of Y3Al5O12 were investigated by firstprinciples calculations and experimental investigations. The calculated hardness (18.2 GPa), the elastic stiffness (C11 ¼ 309.5 GPa; C12 ¼ 106.8 GPa; C14 ¼ 103.7 GPa), bulk modulus (174.4 GPa), Yong’s modulus (254.7 GPa) and Poisson ratio (0.256) are in good agreement with the experimental results. The hardness of YAG is mainly attributed to AleO bonds, especially for AlteteO bonds with high Mulliken populations and short bond length. The elastic anisotropy was also discussed in this paper. The elastic stiffness of YAG is inclined to isotropic from its Zener anisotropy ratio (1.02) and its universal elastic anisotropic index (6.3E-4). The Debye temperature of YAG is 847.6 K, while according to the experimental velocity, it is 822.4 K. They keep in good agreement with each other.

½100V1 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi C11 =r

(11)

½010 Vt1 ¼ ½001 Vt2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi C44 =r

(12)

It is evident that the material with smaller density r and larger elastic constant Cij has larger sound velocities. The calculated and measured sound velocities for YAG are shown in Table 4. The Debye temperatures,QD , for YAG was calculated from the average sound velocity, vm, based on the following equations [26]

QD ¼



  h 3n NA r 1=3 vm kB 4p M

(13)

Where h is the Plank’s constant, kB is the Boltzmann’s constant, n is the number of atoms in the molecule, NA is the Avogadro’s number, r is the density, and M is the molecular weight. The average sound velocity is defined as

" vm ¼

1 2 1 þ 3 v3 v3t l

!#1

Acknowledgment This research is supported by National Science Foundation of China (50990302). References

3

(14)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

nl ¼ pðB þ 4G=3Þ=r ffiffiffiffiffiffiffiffi ns ¼ G=r

(15)

where B and G are isothermal bulk modulus and shear modulus. As shown in Table 4, the Debye temperature of YAG from calculated elastic constants is 847.6 K, while based on the experimental velocity, it is 822.4 K. They keep in good agreement with each other. Table 4 Longitudinal (vl), transverse (vt) and average (vm) sound waves velocity (m/s), Debye temperature (QD , K) of Y3Al5O12. Source

r (g/cm3)

vl (m/s)

vt (m/s)

vm (m/s)

QD (K)

Experiment Calculation

4.546 4.55

8485 8236

4850 4704

6293 6105

847.6 822.4

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