# Influence of growth process and crystal defects on sapphire brittleness

Jessica Muzy, Marc Fivel, Serge Labor, Denis Guignier, José Baruchel, Thu

Nhi Tran Thi Caliste, Vitali Nagirnyi, Ivo Romet, Thierry Duffar

► To cite this version:

Jessica Muzy, Marc Fivel, Serge Labor, Denis Guignier, José Baruchel, et al.. Influence of growth process and crystal defects on sapphire brittleness. Journal of Crystal Growth, 2023, 618, pp.127327. 10.1016/j.jcrysgro.2023.127327. hal-04132412

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Submitted on 19 Jun 2023

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# Influence of growth process and crystal defects on sapphire brittleness

Jessica Muzy<sup>a</sup>, Marc Fivel<sup>b</sup>, Serge Labor<sup>a</sup>, Denis Guignier<sup>a</sup>, José Baruchel<sup>c</sup>, Thu Nhi Tran

Thi Caliste<sup>c</sup>, Vitali Nagirnyi<sup>d</sup>, Ivo Romet<sup>d,e</sup>, Thierry Duffar<sup>b*</sup>

<sup>a</sup> RSA le rubis SA, BP16, 38560 Jarrie, France

<sup>b</sup> Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMAP, Grenoble, France

<sup>c</sup> ESRF, BP 220, 38043 Grenoble, France

<sup>d</sup> Institute of Physics, University of Tartu, W. Ostwald Str. 1, 50411 Tartu, Estonia

<sup>e</sup> Institute of Physics PAS, Lotnikow Av. 32/46, 02-668 Warsaw, Poland

* Corresponding author, thierry.duffar@grenoble-inp.fr

## Abstract

Brittleness of sapphire has been studied by four point bending and ball on three ball tests. A significantly larger flexural stress was observed for crystals grown by the Verneuil process compared to EFG (Edge-defined Film-fed Growth) crystals. Crystallographic defects were characterized by X-ray topography and it was shown that subgrain boundaries found in Verneuil crystals do not impact the fracture behavior. Larger amounts of basal dislocations in Verneuil boules appeared to account for the differences in brittleness. The dislocation densities have been related to the temperature fields experienced by the crystals during the growth processes. Characterization of point defects by thermo-stimulated luminescence revealed that they are more numerous in Verneuil crystals, especially after annealing, and that they pin dislocations, contributing to the mechanical response of the crystal.

## Keywords

B1. Sapphire; A1. Brittleness; A2. Verneuil; A2. EFG; A1. Dislocations; A1. Point Defects.
# 1 Introduction

Synthetic sapphire, Al₂O₃, pure or appropriately doped, is currently used in countless advanced technology applications in the fields of optics [1], lasers [2], microelectronics [3], infrared and ultraviolet windows, and even surgery [4]. In parallel, an important production concerns older uses linked to its high hardness: jewelry, clock making, scratch-proof windows and screens, including clock glasses, and various hard mechanical pieces such as wire guides and pulleys, or ultra-high pressure water nozzles [5]. Directly in all these mechanical utilizations, and indirectly in the advanced technology uses, understanding the brittleness behavior of sapphire is of importance.

Sapphire is a rhombohedral crystal, belonging to the R$\overline{3}$c space group, usually described in the hexagonal triple cell [5]. Oxygen anions are distributed in a compact hexagonal structure, with aluminum cations located in two over three octahedral sites. This structure, associated to strong, mainly ionic, bonds, results in a hard, brittle material, in which, at ambient temperature, mechanical energy is essentially dissipated by rupture. The elastic constants of sapphire [6] show that it is a moderately anisotropic material, with Young modulus variations limited to 20% (anisotropy factor A=0.88). At room temperature, the cleavage plane is (11$\overline{2}$0) [7]. The published mechanical properties are listed in Table 1.

<table><caption>Table 1: Mechanical properties of sapphire at ambient temperature. When the growth process is known, it is mentioned after the reference: V for Verneuil, E for Edge-defined Film-fed Growth (EFG), K for Kyropoulos, H for Heat Exchanger Method (HEM). A is for annealed samples and C for calculated values.</caption><thead><tr><th>Property</th><th>c axis</th><th>a axis</th><th>m axis</th><th>r axis</th></tr></thead><tbody><tr><td></td><td>&lt; 0001 &gt;</td><td>&lt; 1010 &gt;</td><td>&lt; 1120 &gt;</td><td>&lt; 1102 &gt;</td></tr></tbody></table>
<table><tbody><tr><td>Vickers micro-hardness<br/>[kg·mm<sup>-2</sup>]</td><td>2878</td><td>[8] E, A</td><td>2212</td><td>[8] E, A</td><td>2212</td><td>[8] E, A</td><td></td><td></td></tr><tr><td rowspan="6">Young modulus<br/>[GPa]</td><td>450 ± 1</td><td>[9] V</td><td></td><td></td><td>410 ± 1</td><td>[9] V</td><td>340 ± 1</td><td>[9] K</td></tr><tr><td>454 ± 1</td><td>[9] K</td><td>422 ± 1</td><td>[9] K</td><td>421 ± 5</td><td>[9] K</td><td>345 ± 2</td><td>[10] H</td></tr><tr><td>455</td><td>[8] E, A</td><td>423 ± 1</td><td>[9] V</td><td>424</td><td>[7] E</td><td>379 ± 1</td><td>[9] V</td></tr><tr><td>456,5</td><td>[11] C</td><td>430 ± 5</td><td>[10] H</td><td>426 ± 4</td><td>[10] H</td><td>386.1</td><td>[11] C</td></tr><tr><td>460</td><td>[7] E</td><td>431.2</td><td>[11] C</td><td>431.2</td><td>[11] C</td><td></td><td></td></tr><tr><td>465</td><td>[10]</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="6">Average ultimate stress<br/>(4 point bending test)<br/>[MPa]</td><td>703</td><td>[12]</td><td></td><td></td><td>350</td><td>[8] E, A</td><td>427</td><td>[12]</td></tr><tr><td>726 ± 211</td><td>[9] K</td><td>350</td><td>[8] E, A</td><td>439 ± 65</td><td>[9] K</td><td>452 ± 75</td><td>[9] K</td></tr><tr><td>930</td><td>[8] E, A</td><td>489 ± 53</td><td>[9] K</td><td>501 ± 316</td><td>[13] H</td><td>585 ± 141</td><td>[9] V</td></tr><tr><td>1055 ± 359</td><td>[13] H</td><td>591 ± 156</td><td>[9] V</td><td>703 ± 232</td><td>[13] H</td><td>595</td><td>[12]</td></tr><tr><td>1061</td><td>[12]</td><td>784 ± 275</td><td>[13] H</td><td>726 ± 254</td><td>[9] V</td><td></td><td></td></tr><tr><td>1145 ± 229</td><td>[9] V</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">Weibull stress [MPa];<br/>Weibull exponent<br/>(4 points flexure)</td><td>725 ± 45; 6±2</td><td>[14] H</td><td></td><td></td><td></td><td></td><td>483; 6.60</td><td>[9] K</td></tr><tr><td>812; 3.42</td><td>[9] K</td><td>514; 9.45</td><td>[9] K</td><td>467; 7.30</td><td>[9] K</td><td>545 ± 14; 4.1</td><td>[12]</td></tr><tr><td>946 ± 166; 3.4</td><td>[12]</td><td>656; 3.70</td><td>[9] V</td><td>815; 3.16</td><td>[9] V</td><td>558 ± 54; 4.1</td><td>[12]</td></tr><tr><td>1001 ± 78; 3.4</td><td>[12]</td><td></td><td></td><td></td><td></td><td>642; 4.34</td><td>[9] V</td></tr><tr><td rowspan="5">Fracture toughness<br/>[MPa·m<sup>-1/2</sup>]</td><td>0.026</td><td>[15]</td><td>2.43 ± 0,26</td><td>[16]</td><td>2.51</td><td>[11] C</td><td>2.15</td><td>[11] C</td></tr><tr><td>4.54 ± 0.32</td><td>[16]</td><td>2.51</td><td>[11] C</td><td>2.53 ± 0.06</td><td>[9] K</td><td>2.38 ± 0.14</td><td>[16]</td></tr><tr><td>5.62</td><td>[17] C</td><td>2.73 ± 0.13</td><td>[9] K</td><td>3.14 ± 0.30</td><td>[17]</td><td>2.67</td><td>[17] C</td></tr><tr><td>6.04</td><td>[11] C</td><td>2.84</td><td>[17] C</td><td>5.19 ± 0.13</td><td>[9] V</td><td></td><td></td></tr><tr><td>7.0</td><td>[18] V</td><td>3.05 ± 0.32</td><td>[9] V</td><td></td><td></td><td></td><td></td></tr><tr><td>Compressive strength<br/>[MPa]</td><td>1972 ± 457</td><td>[13] H</td><td>2002 ± 293</td><td>[13] H</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="3">Surface energy<br/>[J·m<sup>-2</sup>]</td><td>21.54</td><td>[10]</td><td>8.1±0.8</td><td>[9] K, C</td><td>7.0 ± 0.3</td><td>[9] K, C</td><td>6</td><td>[11] C</td></tr><tr><td>40</td><td>[11] C</td><td>7.3</td><td>[11] C</td><td>7.3</td><td>[11] C</td><td>6.0</td><td>[19] V</td></tr><tr><td>&gt; 40</td><td>[20] V</td><td>10.2 ± 2.1</td><td>[9] V, C</td><td>7.3</td><td>[8] [19] V</td><td>7.68</td><td>[10]</td></tr></tbody></table>
<table><tr><td></td><td>50.7 [18] V</td><td></td><td>11.43 [10] &gt; 7.8 [18] V</td></tr><tr><td></td><td></td><td>30.2 ± 1.5 [9] V,C</td><td></td></tr></table>

There are large dispersions in the values, due to crystals of various origins, grown by different processes, and characterized by different techniques. Generally speaking, Verneuil crystals show higher mechanical properties, such as stress at brittleness, toughness or surface energy, compared to crystals grown by other methods, but the various mechanical properties are differently impacted by the process. For example, Graça [9] compared crystals grown by the Verneuil and Kyropoulos methods. The Young modulus does not depend on growth process, but the brittleness stress along the $\vec{c}$ axis is 50% larger in the case of Verneuil crystals, should it be given as a mean value or as a Weibull statistic. Klein [12] measured the mean brittleness stress of samples coming from two providers but did not mention the growth method. There was a variation of 30% from one provider to the other. However, when the same measurements are presented as a Weibull statistic, the difference between the two providers is reduced to 10%. Gurjiyants [8] compared as-grown EFG crystals and crystals annealed after growth, 4 hours at 1900°C under vacuum. Crystal hardness is strongly impacted by annealing: from 2157 kg·mm⁻² for basal plane and 1424 kg·mm⁻² for prismatic planes, it increases to, respectively, 2878 kg·mm⁻² and 2212 kg·mm⁻² after annealing.

Another effect concerns the crystallographic orientation of the samples. In the case of crystals grown by the heat exchanger method, Schmid and Harris [13] found a 40% increase of the 4 point bending brittleness stress, between samples loaded along the $\vec{m}$ axis. The difference was that the larger face of the sample was $\vec{c}$-oriented in one case and $\vec{a}$-oriented in the other one. Finally, the sample machining, polishing and handling impacts the results, as shown by D. Black [14] with measurements performed jointly with X-ray topography. Annealing can be used in order to improve the surface quality and the mechanical properties [21]. Chemical polishing allowed increasing the mechanical strength by a factor two [12]. It was even shown that the
polishing direction [22], the long exposure to solvents [23], or the measurement atmosphere
[10] can impact the results.

In conclusion, the mechanical, and especially brittleness, properties of sapphire single crystals
are severely impacted by the crystal producer and growth method, including post-growth
annealing, by the crystallographic orientation and by the preparation and handling of the
samples. It should further be noted the strong effect of temperature [8; 24; 25] and dopant [26
- 28] but these parameters are not taken into account in this contribution, that focuses on
commercial white sapphire at ambient temperature. Considering the wide dispersion of the
results, several authors performed statistical measurements in order to be able to conclude to
significant effects.

For a given process, it is not clear which parameter impacts significantly the crystal brittleness.
A first objective of the present work was to study the effect of some growth parameters of
importance in the processes of the industrial partner, especially with respect to the production
of clock glasses. As recommended by several authors, reproducible polishing, handling and
sample tracking, as well as a statistical approach [12], will be used, as explained in the part 2.
Two flexural tests have been used: classical 4-point bending (4PB) and ball on 3 balls (B3B).
These tests are complemented by nano-indentation experiments in order to analyze the onset of
plasticity. Results concerning crystals grown by the Verneuil and by the EFG methods will be
presented in the third part.

It also appears that no publication succeeded to make a link between the rupture properties of
sapphire and the crystal defects. As they are obviously inherited from the growth process, this
is possibly the missing link between process and mechanical property, likely to contribute to
the understanding of the observed dispersion. Therefore, the second objective of this paper is
to characterize the studied crystals in terms of crystallographic defects such as dislocations,
sub-grain boundaries or point defects (section 4) and investigate the possible correlations with
the mechanical property variations and growth process (section 5). Since the samples containing
macro and micro-bubbles are systematically removed from the production by the industrial
partner of this study, this type of defect, at least as far as it can be detected, will not be
investigated here.

## 2 Sampling

The 4PB tests require elongated parallelograms, and the B3B tests use circular disks. Keeping in mind the above mentioned dispersion sources, great care has been provided to all the steps preceding the flexural tests, in order to assess the reliability of the results: growth, cutting, polishing, quality inspection, with enough samples for assessing statistic validity. Double check traceability has been systematically applied all along these steps.

### 2-1 Growth processes

The Verneuil and EFG methods are both used in order to produce clock glasses. Crystals were taken directly from the industrial production lines where they have been grown under reproducible conditions, with only one process parameter changing from one sample series to another. Several sample series have been tested in order to study the effect of process parameters on the rupture behavior, such as furnace aging, growth rate and others, of interest for the industrial partner. The raw powder used for the Verneuil production was slightly doped with Ti (10 ppm), Fe (10 ppm) and Cr (2 ppm). Values are lower in case of EFG, because the raw material is a mixture of Verneuil pieces and undoped alumina powder.

The industrial Verneuil process is very similar to the initial process [29] but has been improved in order to produce larger crystals: cylindrical samples as big as 35 mm in diameter and 150 mm long, also called ‘boules’. It uses sets of identical ceramic furnaces which receive the same gas flows (H₂ and O₂) and have the same seed pulling rate. The growth of each individual crystal is controlled by the powder flowrate provided to the furnace. After growth, the crystals are
annealed at 1800°C for 48 hours under nitrogen in order to release the residual stresses. As-
grown boules invariably break during cutting, so that it has only been possible to perform
flexural tests on annealed samples. Clock glasses are cut perpendicularly to the axis of the
cylindrical boules.

The EFG process [30] uses a multi-die crucible in order to pull simultaneously several tens of plates. No annealing is needed and clock glasses are cored from the plates. Pulling rates is of the order of 10 µm.s⁻¹.

**2-2 Cutting and polishing**

For the 4 point bending test, the French standard (NF EN 843-1: 2007) recommends two sample
sizes 2 × 2,5 × 25 mm³ and 3 × 4 × 45 mm³. As some EFG plates were thinner than 3 mm after
polishing, the smaller size was chosen. 4PB tests have been performed with Verneuil samples
of both sizes and have shown a significant difference, but included in the dispersion revealed
in table 1. Consequently, the present results cannot be quantitatively compared to other
literature results. For the B3B test, according to recommendations given in [31-35], disks of 2
mm in thickness and 32 to 35 mm in diameter were cut.

In the Verneuil process, the easiest growth and better crystal quality are obtained when the
growth axis is oriented at 13° from the *a* axis, in direction of the *c* axis. This specific orientation
remains for the clock glass disks which are cut perpendicularly to the pulling direction.
Consequently, in order to get similar glasses, the EFG plates are pulled along the *m* axis, with
the same 13° angle between the *a* axis and the normal to the plate. The long side of the 4PB
samples was oriented along the *m* axis, which gives a Schmid factor of 0.46 for the two others
*a* and *m* planes. Theoretically, this specific orientation allows to access the yield stress at the
onset of plasticity, in the case dislocations are present and could accommodate the applied
loading. It also enables estimation of the flexural stress from the applied force measured when
the specimen breaks.
The dimensions and orientations of the 4PB and B3B samples were as shown on the figure 1.

It should be noted that the 13° angle did not exist for some EFG plates, where $\vec{a}$ and $\vec{c}$ axes
were perpendicular to the faces. Finite element simulation of anisotropy effects allowed to
determine correction factors in order to get rid of this difference [36]. Depending on the
machined crystal dimensions, the diameter of the B3B samples varied between 32.7 and 35 mm.

Comparison tests have shown that this does not significantly impact the measured flexural
strength.

<figure><img src="image_2.png" /><figcaption>Figure 1: Dimensions and crystallographic orientations of the samples used for the 4PB (left) and B3B (right) flexural tests.</figcaption></figure>

Before cutting, each crystal was measured, in order to remove those out of dimension, and
carefully examined in order to exclude those showing bubbles. As the end (top) of the Verneuil
crystals always present bubbles of various sizes and configurations, they were systematically
cut out. Samples were cut with a diamond wire saw or diamond coring tools. X-ray diffraction
was used for the orientation selection in order to limit the maximal misorientation to 1°.

The surface condition is of the uppermost importance in flexural tests, as any scratch might be
the origin of a crack. Hence, the two faces of the disks and the long faces of the parallelepipeds
were polished. This was performed by specialized external companies with strict and
reproducible procedures. Surfaces were first grinded with boron carbide in order to remove
cutting defects and homogenize the dimensions of all samples. Then diamond polishing was
performed in the usual way of decreasing powder size. The final mechano-chemical polishing step used a basic (pH>12) colloidal silica solution.

Atomic Force Microscopy analyses gave a roughness of 1 to 2 nm for the faces of 4PB Verneuil samples, with occasional scratches of 12 nm deep. Scratches or chips, up to 1 µm, have also been found on the edges. Three different polishing series were used for the EFG 4PB samples; statistical analyses of the corresponding flexural stresses did not show any significant difference. This shows that, in spite of the effect of surface or edge scratches, the results can be used for comparison purpose. On the B3B disks, the roughness was lower than 1 nm, without any scratch. This is equivalent to the typical roughness of sapphire substrates used for microelectronics [37].

## 2-3 Inspection

Before test, each polished sample was examined in order to limit, or understand, the variation of results. The 4PB sample dimensions were precisely measured (± 1 µm). In a series, the dispersion was around 10 µm, that corresponds to a 2% error on the stress calculated from these dimensions. The thickness of the B3B disks was measured (± 5 µm) which contributes to 1% error on the stress calculation, while a 1 mm error on the diameter contributes only to 0.5% [32].

Visual inspection under laser beam revealed internal bubbles and veils (which are sub-grain boundaries decorated with microbubbles) and surface damages such as scratches or small breaks on ridges. Samples presenting veils were subject to a specific rupture analysis. All 4PB tests have been performed with samples without any visible defect. Some B3B tests series did not have 30 perfect samples. In this case, defective samples were used, but taking care that defects such as surface scratches or bubbles were placed on the side of the sample which experienced compression.
The crystallographic orientation of all EFG samples has been measured with an X-ray goniometer. This was not possible in the case of Verneuil samples because they always present sub-grains, that can reach a few degrees of misorientation. Their structure was characterized by observation under crossed polarizers. The sub-grain structure becomes progressively more complex from the beginning (bottom) to the end (top) of the boule, as can be seen on figure 2.

<figure><img src="image_3.png" /><figcaption>Figure 2: Picture under crossed polarizers of disks from the beginning (left) and end (right) of a same Verneuil boule, 35 mm in diameter.</figcaption></figure>

## 2-4 Statistics

Some 2000 samples have been machined, among which 1300 have been broken. Such large numbers are due to the analysis of the effect of several industrial process parameters, each series of samples containing at least 30 broken pieces. In the following text, a “significant” or “non-significant” result means that a statistical treatment has been performed, which includes:

- Systematic rejection of outlier values, in order to avoid result dispersion due to undetected defects. The Dixon test [38] was used, with a 10 % risk of wrongly rejecting a value. In case of merging several series, the Tukey's rule was used [39]. In all cases, the rejected values were not taken into account in subsequent statistical analyses but they are shown on the graphs, with a probability of 0 or 1.

- Gauss statistical analysis. The parameters, mean value $\mu$ and standard deviation $Std$ were calculated. The standard deviation is given as “± ...”. Weibull distribution parameters, $m$ and $\sigma_0$, were also computed in some cases.
- Kolmogorov-Smirnov test, in order to compare two sample series, with a confidence level of 10 % [40].

# 3 Mechanical properties

## 3-1 Methods

### 3-1-1 Four-point bending tests

This mono-axial method is commonly used because it applies a pure flexure loading condition, allowing exact calculations of the stresses and properties. The equipment has been designed in agreement with the French standard (NF EN 843-1: 2007) and can be seen in figure 3-a. The upper alumina rolls were free to move inwards and the lower alumina rolls free to move outwards. Each roll was also free to rotate around an axis parallel to the sample in order to match the parallelism with the compression platen. Metallic pieces were machined in 35NCD61 or 45NCD16 steels, quenched in order to reach a hardness larger than 50HRC, then resurfaced.

The set-up was placed in the jaws of a tensile testing machine with a load cell of 2 kN.

<figure><img src="image_4.png" /></figure>
Figure 3: Test block setups used for: a) 4PB test; b) B3B tests (the centering tool, which allows accurate positioning of the balls, is moved down before the test).

### 3-1-2 Ball on three balls

This is a bi-axial flexure test, likely to be more representative of the mechanical behavior of usual mechanical pieces [32]. As the stress is applied in the center of the disk, the detrimental effect of edge defects, such as cracks or chips, is avoided. Also, because all radial directions are involved, the effect of internal defects is better evidenced. Klein et al. [41; 42] found that flexural stresses of sapphire are less dependent on the provider when measured by a bi-axial method. The ring on ring test has been used by Fisher [24] in the case of sapphire but appeared difficult to use because of drastic constraints on the ring positions. Therefore, the ball on three balls test has been chosen in this study, as it is easier to perform [31; 32]. A drawback is that there is no analytical solution derived to compute the stresses, because the stress at the contact points is quite complex [43]. Flexural stress was derived according to the method presented in [31], taking a Poisson coefficient of 0.30 for sapphire. A finite element study has been performed in order to quantify the anisotropy effect due to the specific orientation of the samples [36]. The equipment is shown in figure 3-b. All 100C6 steel balls had the same diameter of 17 mm and the three upper ones were in contact with each other, so that they can only move outwards. Other metallic pieces have been produced in the same way as for the 4PB test and the same testing machine was used, with a load cell of 10 kN.

## 3-2 Results

### 3-2-1 Four-point bending

A first series of experiments was performed in order to check the effect of misorientations on the results. No significant effect was found for variations of a few degrees out of the sample theoretical axes. Therefore, small misorientations among the samples can be neglected. Observation of breaking patterns allowed to find the incipient crack positions. The crack nucleated at the edge in 80 % of the cases and on the face for the other 20 %. 55 % initiated in
the center of the sample and 45% at the tip of an upper roll. No significant difference was found for the flexural stresses measured in the different cases. A significant correlation was found between the number of broken pieces and the flexural stress.

A Gom Aramis Adjustable camera has been used in order to measure the deformation of three speckled Verneuil samples during 4PB tests. Analyses of the movements of the loading points gave a Young modulus of 470±31 GPa. This value is higher than usually measured for *a*-oriented samples (see Table 1), possibly because the Verneuil samples were misoriented by 13° from this axis.

<figure><img src="image_5.png" /><figcaption>Figure 4: Breaking probability as function of stress in 4PB (a) and B3B (b) tests of Verneuil and EFG samples.</figcaption></figure>

As can be seen in figure 4-a, where all measurements are represented for both processes, and in table 2, the flexural stresses measured for EFG samples (450 ± 83 MPa) are lower than for Verneuil samples (490 ± 94 MPa). The difference is significant, which is also the case for the Weibull parameters. Comparison with Table 1 shows that the obtained values are consistent with measurements performed on samples with similar crystallographic orientations.

Table 2: Gauss (μ, Std) and Weibull (σ₀, m) parameters of the flexural stresses measured by the 4PB and B3B methods on samples produced by Verneuil and EFG processes.
<table><thead><tr><th rowspan="2"></th><th colspan="2">Verneuil</th><th colspan="2">EFG</th></tr><tr><th>4PB</th><th>B3B</th><th>4PB</th><th>B3B</th></tr></thead><tbody><tr><th>μ (MPa)</th><td>490</td><td>1840</td><td>450</td><td>1740</td></tr><tr><th>Std (MPa)</th><td>94</td><td>630</td><td>83</td><td>720</td></tr><tr><th>σ₀ (MPa)</th><td>520</td><td>2040</td><td>480</td><td>1940</td></tr><tr><th>m</th><td>6.4</td><td>3.5</td><td>6.6</td><td>3.2</td></tr></tbody></table>

### 3-2-2 Ball on three balls

The first series of tests has been devoted to the positions of the balls with respect to the crystallographic axes. No significant difference was found between the cases where a ball was positioned on the projection of the $\vec{c}$ axis or at 30°. However, further tests were performed with a ball on the projection of the $\vec{c}$ axis.

The broken disks usually show three diametric cracks. They correspond to a *m* plane, parallel to the $\vec{c}$ axis, on 85 % of the disks, to a *r* plane oriented at +65° of this *m* plane, on 85 % of the disks, and to another *r* plane, at -45° of the *m* plane, for 92% of the disks.

As for the 4PB test, the B3B test revealed a significant difference between the flexural stresses of the Verneuil (1840 ± 630 MPa) and EFG (1740 ± 720 MPa) samples, as shown in figure 4-b and Table 2. The value obtained for the flexural stress is much higher when measured by the B3B test (around 1800 MPa), compared to the values obtained by the 4PB test (around 500 MPa). This difference is attributed to the sample surface quality: the 4PB samples show scratches on the faces and on the edges, that the disk surfaces did not show at all. Assuming that the flexural stress is correlated to the toughness, $K_{Ic} = \sigma\sqrt{a}$, with *a* the depth of cracks, the factor four between the flexural stresses is consistent with the factor >12 in the measured sample roughness.
There was a significant difference between the top and the bottom of the Verneuil boules. However, as already mentioned, numerous boules show veils at the end of their crystallization (i.e. at the top). Figure 5 shows the distribution of the flexural stress for the disks cut at the top of the boules, with distinction between the clean boules and those showing veils: the difference is significant. Whereas the difference between top and bottom disappears when only clean boules are taken into account, showing that veils are responsible for the observed difference.

<figure><img src="image_6.png" /><figcaption>Figure 5: B3B flexural stress distributions for samples cut at the top of Verneuil boules.</figcaption></figure>

### 3-3 Nano-indentation

Nano-indentation can be used in order to measure Young modulus [15] and to study dislocation generation and movement in sapphire [44], through the analysis of pop-in events. By this way, it was shown that plasticity occurs at lower energy for low quality crystals than for crystals with high quality. This is because, in this second case, it is necessary to generate dislocations whereas they are already present in the low quality samples [23]. Such experiments should take into account the crystallographic orientation of the sample surface [45].
Thirty to sixty nano-indentation tests were performed on samples corresponding to the various process parameter combinations. The equipment used for these characterizations is a UNHT3© model from the Anton PAAR company equipped with a 90° conical-spherical indenter tip of 250 nm radius.

Pop-in events have been systematically recorded in order to analyze the occurrence of plasticity. A pop-in event is defined as the strain burst induced by a sudden and brief plastic event. It can easily be detected from the derivative of the penetration depth signal. The tip radius was measured before and after the tests using a fit of the Hertzian regime on fused silica calibrated samples. It could be concluded that the radius did not change during the tests. All tests were performed at a load rate of 20 mN·min⁻¹, with a waiting time of 10 s at the maximal load of 20 mN.

Figure 6 shows that the elastic part of the experimental plots fits well to the theoretical Hertz curve corresponding to a Young modulus of 470 GPa. This value is equal to the value obtained from 4PB tests. However, several EFG samples were also included, that did not show the 13° misorientation. Therefore, this misorientation cannot be considered to impact the measure, and the studied crystals, Verneuil as well as EFG, have a higher Young modulus, compared to the values found in the literature: 430 GPa along the $\bar{a}$ axis. As the doping level is small, around 20 ppm, this difference remains unexplained.

<figure><img src="image_7.png" /></figure>
Figure 6: Results of nine nano-indentation measurements, corresponding to various process parameters, compared to the theoretical Hertz plot (in thick blue). The pop-in events are detected as the burst in the penetration signal, i.e. the horizontal excursions.

The first pop-in corresponds to the first occurrence of plasticity, i.e. the initial movement of dislocations. The corresponding force depends on the density of preexisting dislocations and on the surface orientation (ā or 13° from a, depending on the sample). Mean values are 6.9 mN for Verneuil and 4.2 ± 0.13 mN for EFG, however the dispersion of the force values for the Verneuil samples is large, so that it is not possible to conclude about a significant effect of growth process.

# 4 Crystal defects

## 4-1 Sub-grain boundaries

As shown in figure 2, sub-grain boundaries can easily be observed between crossed polarizers. Polarizers are oriented in such a way that sub-grains with a vertical (on the picture) c axis appear in dark and other sub-grains are clearer and clearer when the misorientation increases, with a sensibility of around 0.1°. However, the technique is not quantitative. Sub-grain boundaries only exist in Verneuil boules, with vertical boundaries in the pictures, showing that they are close to *m* planes. Misorientations are lower than a few degrees. As already published [46], disks cut at the top (end of growth) of the boules always show numerous small sub-grains, while disks cut at the bottom also show sub-grains but larger and less misoriented.

In order to measure the grain misorientations, white beam X-ray topography (wavelengths from 20 pm to 200 pm) was performed at the BM05 beamline of the European Synchrotron Radiation Facility (ESRF). Sections of the sample were obtained with a 50 µm thick and 35 mm wide beam. As can be seen in figure 7-a, rotations of grains result in a displacement of the grain image on the diffracted picture: $e_x$ for a rotation around an axis lying at the intersection of the reflecting planes and the diffraction plane and $e_y$ for a rotation around an axis perpendicular to
the diffraction plane. The possible rotation around the axis perpendicular to the family of reflecting planes is not visible in this experiment. Taking into account geometrical corrections depending on the diffraction angle $\theta_D$, and usual small angle approximation, the measured misorientation angle $\alpha$ is given by:

$$ \alpha \approx \frac{1}{2d} \left( e_x \frac{\cos\theta_D}{\sin(\frac{\theta_D}{2})} + e_y \cos^2\theta_D \right), $$

where $d$ is the distance between sample and film (300 mm). Figure 7-c shows on the left a disk where grains are observed between crossed polarizers. The picture obtained for the section diffraction of the (1017) plane is shown on the right. The images of the grains are separated, and distorted due to the presence of dislocations (figure 7-b).

<figure><img src="image_8.png" /><figcaption>Figure 7: White beam topography: a) Several grain diffractions; b) Grain distortion due to dislocations; c) Example on a Verneuil disk (left, between polarizers) and its (1017) diffracted 50 µm thick section (right).</figcaption></figure>
Measurements on Verneuil crystals show a significant difference between the grain misorientations at the top (2.3 ± 2.9 °) and at the bottom (0.7 ± 0.6 °) of the boules. However, the difference in flexural stress was not significant, showing that it is not impacted by sub-grain boundaries.

Thirteen EFG plates have been studied by transmission white beam topography (8 mm x 35 mm beam). Among the 30 pictures obtained, only one has shown two sub-grain boundaries, with a misorientation lower than 0.1°.

## 4-2 Dislocations

The inner-grain distortion due to dislocations can also be measured on X-ray topographs. Assuming that each dislocation introduces a distortion equal to its Burgers vector $\vec{b}$, the density of dislocations needed to account for the total distortion can be estimated by using the formulae: $\rho = \frac{\Delta\theta}{||\vec{b}|| \cdot L}$. L is the length on which the distortion $\Delta\theta$ occurs (see figure 7-b). For both prismatic and basal glide systems, which are activated at high temperature during growth, the Burgers vector is the cell parameter $a=4.76$ Å [47]. In the case of Verneuil crystals, the density of dislocations at the top of the boules, where there are numerous sub-grains, appeared lower (1.7 $10^9$ m$^{-2}$). It was higher (2.2 $10^9$ m$^{-2}$) at the bottom where few sub-grains were observed. However, due to large dispersions in the measured densities, the difference is not statistically significant.

EFG samples show much lower dislocation densities, which were barely measurable on the topographs. In order to get more quantitative information on the dislocation patterns, Rocking Curve Imaging (RCI), which allows measurement of the spatial distribution of local lattice distortions [48], was also performed on the ESRF-BM05 beam line, in both projection [49] and section [50] modes. For each diffraction peak, this technique provides three maps. The peak intensity map gives results similar to the topography image, as it shows the amount of diffracted
beam. The peak position map gives local lattice rotations. The peak FWHM (Full Width at Half
Maximum) map shows the local lattice distortions.

Projection maps were first performed, the studied area being 10.2×10.2 mm², with 5×5 µm² pixels. In order to study the distribution of the defects in the sample thickness, multi-sections were obtained, after passing the X-ray beam through a gold foil with 10 µm slits, 400 µm apart. In that case the samples were thinned to 1 mm, in order to avoid overlapping of the section images on the detector. The {0006} and {1120} diffractions have been chosen for the analysis of the basal plane and prismatic planes, respectively, because they allow a clear separation of the multi-sections. The beam energy was 20 keV.

Figure 8 shows several RCI projection maps, and a multi-section intensity map of the same area, of a 1 mm thick sample cut in the center of an 8 mm thick EFG plate. On the {1120} FWHM map, dislocations are visible in red and are aligned along gliding planes corresponding to the basal plane. Counting the dislocations on this map, with an estimation of 200 and 300 in the gliding planes, gives a dislocation density of 1. 10⁷ m⁻². They cannot be seen on the {0006} diffraction, meaning that they are in extinction condition for the basal plane, i.e. their Burgers vectors are contained in this plane. This suggests that the basal glide systems with a ⃗a Burgers vector, which is preferentially activated above 700°C, produced the dislocations. The zoom in the middle of the {1120} map shows that most dislocations are arranged in pairs, or in elliptical sections. The 5 µm pixel size does not allow to distinguish partial dislocations. Then it is hypothesized that dislocation loops are formed during the crystal growth; some loops being cut by the crystal surface. The intensity section maps also allow estimations of the dislocation density. An interesting feature is that in some parts of the sections, horizontal Kato interference fringes can be seen (enlarged picture of the section map in figure 8-c). Such interferences occur when the crystal distortion is of the order or less than the Darwin width, which accounts for perfect crystals [51]. Figure 9 shows a comparison of (0006) FWHM maps of two samples, 1
mm thick, taken in the center and on the surface, at the same level of an 8 mm thick EFG plate.

It can be seen that the crystal quality is better in the center, compared to the surface of the plate
where the dislocation density is 50% higher.

<figure><img src="image_9.png" /><figcaption>Figure 8: a) {0006} and b) {1120} RCI FWHM maps. c) {0006} Intensity maps of 21 sections (interval 400 µm). The three images represent 10.2×10.2 mm² and are obtained on the same sample.</figcaption></figure>

A Verneuil sample has also been studied by RCI projection. The FWHM (0006) map gave
values ten times larger than for the EFG samples. It was no more possible to observe the defects
individually. However, the dislocation density could be computed with Hirsch formula
[52]: $\rho = \frac{FWHM^2}{9b^2}$. Values ranging from 2. 10$^8$ m$^{-2}$ to 9.10$^9$ m$^{-2}$ have been obtained, consistent with the values obtained from white beam topography.

<figure><img src="image_10.png" /><figcaption>Figure 9: {0006} RCI FWHM maps of two EFG samples taken in the centre and on the surface of the same 8 mm EFG plate. Images are 10.2×10.2 mm$^2$.</figcaption></figure>

### 4-3 Point defects

Schottky defects are energetically favored in sapphire, compared to Frenkel defects, [5]. Hence, Al and O vacancies are more likely to be found. Also, impurities originating from the raw material, crucible or atmosphere can be present in the crystals. It should be noted that growth conditions determine the valence state of a dopant. The EFG crystals are grown in highly reductive atmosphere in Mo crucibles with graphite casing, so only the $Ti^{3+}$ ions can be found in the product [53]. On the contrary, the Verneuil flame becomes oxidizing at a certain moment of the growth process, leading to the appearance of a significant amount of $Ti^{4+}$ ions in the boules [54]. Interstitial hydrogen has also been found in the RSA Verneuil crystals [55]. Hydrogen is likely to produce oxygen vacancies in the crystals [5], giving rise to F-centers according to the following reaction:
$$2 Al^{3+} + 3 O^{2-} + H_2 = 2 Al^{3+} + 2 O^{2-} + (V_0^{\ddot{}} + 2e^{-}) + H_2O.$$

<figure><img src="image_11.png" /><figcaption>Figure 10: (a) TSL glow curve of the top part of an as grown Verneuil sample, measured upon X-irradiation (40 keV, 10 mA) during 5 minutes. (b) TSL spectra measured in two temperature intervals for the same sample during the measuring of the glow curve.</figcaption></figure>

Thermostimulated luminescence (TSL) analyses have been performed for the Verneuil and EFG samples in order to study the role of F⁺ and F centers [56] as well as Ti³⁺ and Cr³⁺ ions in the processes of charge carrier trapping and recombination. The samples were first irradiated by X-rays (40 keV; 10 mA, 5 min.), and then heated from room temperature to 600 K at a linear heating rate of 10 K.min⁻¹. Glow curves were recorded for integral thermostimulated luminescence with a Hamamatsu H8259 photomultiplier. All samples were covered by metal masks of exactly the same dimensions to assure quantitative comparison of TSL intensities for the samples studied. EFG samples demonstrated a single intense glow peak of near a first-order kinetics with the maximum at 435 K and FWHM of 30 K. Verneuil samples show a second-
order kinetics in two peaks at 400 and 435 K with the FWHM of 45 K and 65 K, respectively.

As an example, a glow curve for the top part of an as grown Verneuil sample is shown in figure 10-a. The intensities integrated over the peaks were (2.2 ± 0.7) · 10⁶ counts for EFG samples and (25.5 ± 0.7) · 10⁶ counts for the Verneuil samples, while those of the Verneuil samples, taken before the annealing process, gave a value of (3.5 ± 0.7) · 10⁶ counts.

The setup applied for the TSL studies enabled simultaneous registration of both TSL glow curves and TSL spectra. In the latter case, an ARC Spectra Pro-300i monochromator equipped with a CCD camera was used. According to [57], the TSL spectra in the region of the main peaks are constituted by the F⁺ center emission peaking at 330 nm, F center emission at 410 nm, Cr³⁺ ion emission at 630 nm, and Ti³⁺ ion emission at 730 nm. These emissions were observed with different intensities also in the present studies. As an example, the corresponding TSL spectra measured for the top part of as grown Verneuil sample are shown in figure 10-b. All TSL spectra were recorded exactly in the same conditions with the same metal mask on the sample, allowing a direct comparison of emission intensities. Relative contributions of the above-listed luminescence centers into the TSL of the studied crystals are compared in Table 3.

<table><caption>Table 3: Numbers of detected photons for various defects in EFG and Verneuil samples.</caption><thead><tr><th rowspan="2">Sample</th><th colspan="4">Maximal counts</th></tr><tr><th>F<sup>+</sup></th><th>F</th><th>Cr<sup>3+</sup></th><th>Ti<sup>3+</sup></th></tr></thead><tbody><tr><td>EFG</td><td>Ø</td><td>11 000</td><td>Ø</td><td>2 000</td></tr><tr><td>As grown Verneuil – Bottom</td><td>5 000</td><td>5 000</td><td>15 000</td><td>80 000</td></tr><tr><td>As grown Verneuil – Top</td><td>8 000</td><td>18 000</td><td>40 000</td><td>120 000</td></tr><tr><td>Annealed Verneuil – Bottom</td><td>Ø</td><td>100 000</td><td>5 000</td><td>25 000</td></tr><tr><td>Annealed Verneuil – Top</td><td>Ø</td><td>200 000</td><td>5 000</td><td>20 000</td></tr></tbody></table>

Based on the data presented in Table 3, one can observe that the content of Ti³⁺ ions was low in the EFG samples, while Cr³⁺ was not detected at all. The EFG raw material is a mixture of Verneuil crackles and undoped alumina. Furthermore, dopants are usually segregated at the surfaces of pulled plates [58], from where they are removed by polishing. For these reasons,
the amounts of dopants in the EFG samples may be significantly lower than in the Verneuil
crystals. No F⁺ centers were detected in EFG samples and the content of F centers was relatively
low compared to other studied sample, which can be assigned to the effect of the highly
reducing atmosphere in Mo crucibles with graphite casing, in which the plates were pulled.

The Verneuil samples contain much more Ti³⁺ ions than Cr³⁺ ions. These amounts are in good agreement with the initial dopant concentrations 12 ppm and 2 ppm, respectively. The as grown Verneuil crystals reveal also low amounts of F and F⁺ centers, and correspondingly, of oxygen vacancies. However, after the annealing procedure, the amount of the F center increases remarkably, while the number of F⁺ decreases down to undetectable level. This is a signature of the annealing in highly reducing atmosphere, which is conducted in graphite supports at 1800 °C for 40 hours. The EFG growth takes place in a Mo crucible in less reductive atmosphere [53], whereas the crystals are exposed to the reducing atmosphere for shorter time, compared to the time of Verneuil annealing.

**5 Discussion**

5-1 Defects and mechanical properties

In sapphire, the plastic domain at room temperature is limited to 3% only [59]. Its onset can be
estimated from the first pop-in event of the nanoindentation curve. When the density of
dislocations is low, the force measured at the first pop-in is high. According to the Johnson-
Gilman mechanism [60], this is because it is necessary to nucleate dislocations before their
movement, and this applies in sapphire as well [23]. This nucleation needs energy in order to
form the stacking fault associated to partial dislocations [61]. However, our experiments
showed that Verneuil samples contain a higher dislocation density (about 10⁹ m⁻²) and also a
higher average pop-in force (6.9 mN), compared to lower values in EFG plates (respectively
some 10⁷ m⁻² and 4.2 mN). This means that the pre-existing dislocations in Verneuil crystals
are pinned. According to [62], dopants move towards dislocations, especially Ti³⁺ and Cr³⁺
because they have larger ionic radii (resp. 61 and 62 pm) compared to Al³⁺ (54 pm). In order to move, the dislocation should break this Cotrell atmosphere and this requires a larger force. As already discussed, dopant concentrations are higher in Verneuil and Ti is found as Ti⁴⁺, which adds electrostatic effects to the interaction with dislocations. Another point concerns the very high level of oxygen vacancies in annealed Verneuil boules. At such levels, they can nucleate nano-voids [63], not detected by the inspection and characterization techniques that have been employed, likely to interact with the dislocation motion.

Some EFG samples have been studied by nano-indentation and by RCI. Those showing lower dislocation densities (1. 10⁷ m⁻² compared to 4. 10⁷ m⁻²) also show higher pop-in forces (7mN compared to 4 mN), in agreement with the nucleation hypothesis. At the top of Verneuil boules, a larger number of more disoriented sub-grains is observed and nano-indentation gives lower values for the pop-in force. This suggests that sub-grain boundaries may act as sources of dislocations. This is indeed confirmed by RCI which shows that sub-grain boundaries are constituted of numerous dislocations [36].

Regarding the sapphire rupture, the results confirm that Verneuil crystals break at higher stresses than crystals grown by other processes, as already published [9]. Defect characterization clearly show that this is due to the higher number of dislocations, leading to a more extended plastic regime. The EFG samples also show this tendency: the one with a dislocation density of 1. 10⁷ m⁻² is extracted from a series breaking at 1608 ± 599 MPa, while the one with a dislocation density of 4. 10⁷ m⁻² comes from a series breaking at 1842 ± 792 MPa. Verneuil samples grown under different conditions show also a significant correlation between dislocation density and flexural stress [36]. No correlation has been found between the number or size of grains and the flexural stress. Hence, while sub-grain boundaries are suspected to act as dislocation sources, this is not enough to impact the rupture, maybe because the grain boundaries hamper the dislocation movement on long distances.
## 5-2 Processes and crystal defects

It has been found that dislocations mostly lay in the basal planes. This gliding system being active above 700°C [47], it follows that dislocations have been produced during the growth or the subsequent cooling phase. The crystals are heated through the solid-liquid interface and cooled from their side. This distorted temperature field creates thermo-elastic stresses and dislocation density, proportionally to the temperature gradient at the solid-liquid interface and to the crystal diameter or thickness [64; 65]. The temperature gradients at the interface are similar, 5000 K.m⁻¹, in EFG [66; 67] and Verneuil [68] processes. As the diameter is 35 mm for Verneuil boules and thickness 3 to 8 mm for the EFG plates, this accounts for a factor less than 10 on the dislocation density difference. But the main difference between the two processes is that the EFG plates are slowly cooled in the growing furnace, while the Verneuil boules are subject to a thermal shock, when the flame is suddenly blown out. This suggests that the dislocations, in the Verneuil process, are mainly produced during this cooling step, by multiplication of those produced during the growth. They pile up in sub-grain boundaries, in larger amount at the top of the boule, possibly during the subsequent annealing process.

Numerical simulation of heat transfers and thermo-elastic stresses in multi-die EFG growth gives rather low stresses in the plates, 60 MPa, compared to 200 MPa during Czochralski growth of a 5 cm diameter crystal [67]. Anyhow this value is larger than the yield point at high temperature [47] and dislocations are formed. By symmetry, isotherm curvature, and consequently stress, is low at the plate center and larger towards the external surface, accounting for the worse crystal quality on the surface shown in figure 9.

## 6 Conclusion

In this study, some 1300 samples were broken during four point bending and ball on three ball tests, in order to better understand the fracture behavior of sapphire, with a strong traceability
giving comfortable statistics. The main observed effect concerns the difference between crystals grown by the Verneuil and EFG technics. As sample processing was exactly the same, except a somewhat higher doping in Verneuil, the main difference is linked to the growth process itself.

Among the Verneuil crystals, a decrease of the flexural stress was associated to veils (i.e. sub-grain boundaries decorated by microbubbles) observed at the top of the boules.

Crystallographic defects were characterized by white beam topography and rocking curve imaging. Sub-grain boundaries have been measured in Verneuil samples with 2.3° mean misorientation at the top of the boules. They do not impact the flexural stress. The main difference concerns dislocations, 100 times more in Verneuil. They were shown to result from basal slip in the EFG samples. This is also the most probable scenario in Verneuil but it remains to be confirmed: because there are too many defects, thinner samples should be studied, possibly by TEM [69]. The dislocation densities can be related to the different temperature fields experienced by the crystals during the two growth processes. While the understanding of the heat transfer in Verneuil is still in its infancy, the high level of dislocations is certainly due to the huge thermal shock generated when the flame is blown out.

Point defects have been studied by thermostimulated luminescence. There is a 10 times higher dopant concentration in Verneuil crystals, in which oxygen vacancies increase by a factor 20 during annealing, and even more if it is supposed that some of them precipitate as nano-voids.

The plastic deformation starts at higher stresses in Verneuil samples compared to EFG, while they contain more dislocations. This shows that dislocations are pinned in Verneuil specimen because of point defects. However, the exact mechanism should be studied.

Consequently, it is concluded that a higher dislocation density increases the flexural stress. This is also observed among the Verneuil samples on one side, and among EFG samples on the other side, when process parameters are changed. Therefore, dislocations should be favored in order to improve the mechanical properties of sapphire.
## Acknowledgments

Many people have been involved in the technical aspects of this study. They are warmly acknowledged for their enthusiasm and great work. The 4PB and B3B flexural set-ups were engineered and realized by S. Marsucci, the Bodycote company and the FabMSTIC of the University Grenoble Alpes. C. Josserond, C. Rajon and X. Bataillon helped in their implementation and in solving many technical problems. V. Bouteille worked very efficiently on the optimization of the flexural tests. M. Sylvestre and L. Serre did a great job in the definition of cutting, polishing and in checking the 2000 samples. They were machined and polished by the Ceratec, Surface Technology, Del Fiore Pietri Fine and Saphir industry companies. S. Coindeau helped in the data treatment. Finally, L. Giraud took in charge chemical etching experiments, that remain to assess as a reproducible technique for routine characterization of dislocations. V. Nagirnyi and I. Romet acknowledge the financial support of the Estonian Research Council (project PUT PRG111).

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