Journal of Materials Science

, Volume 47, Issue 13, pp 5086–5096

Dislocation structure at a \( \{ {\overline{1} 2\overline{1} 0} \}/ {\langle} 10\overline{1} 0 {\rangle} \) low-angle tilt grain boundary in LiNbO3

Authors

    • Department of Mechanical & Physical EngineeringOsaka City University
  • Eita Tochigi
    • Institute of Engineering InnovationThe University of Tokyo
  • Jun-nosuke Nakamura
    • Department of Mechanical & Physical EngineeringOsaka City University
  • Ippei Kishida
    • Department of Mechanical & Physical EngineeringOsaka City University
  • Yoshiyuki Yokogawa
    • Department of Mechanical & Physical EngineeringOsaka City University
Article

DOI: 10.1007/s10853-012-6373-7

Cite this article as:
Nakamura, A., Tochigi, E., Nakamura, J. et al. J Mater Sci (2012) 47: 5086. doi:10.1007/s10853-012-6373-7

Abstract

LiNbO3 is a ferroelectric material with a rhombohedral R3c structure at room temperature. A LiNbO3 bicrystal with a \( \{ {\overline{1} 2\overline{1} 0} \}/ {\langle}10\overline{1} 0{\rangle}\) 1° low-angle tilt grain boundary was successfully fabricated by diffusion bonding. The resultant boundary was then investigated using high-resolution TEM. The boundary composed a periodic array of dislocations with \( b = { 1}/ 3{\langle} \overline{1} 2\overline{1} 0{\rangle} \). They dissociated into two partial dislocations by climb. A crystallographic consideration suggests that the Burgers vectors of the partial dislocations should be \( 1/ 3{\langle}01\overline{1} 0{\rangle} \) and \( 1/ 3{\langle}\overline{1} 100{\rangle} \), and a stacking fault on \( \{ {\overline{1} 2\overline{1} 0} \} \) is formed between the two partial dislocations. From the separation distance of a partial dislocation pair, a stacking fault energy on \( \{ {\overline{1} 2\overline{1} 0} \} \) was estimated to be 0.25 J/m2 on the basis of isotropic elasticity theory.

Introduction

LiNbO3 is a widely used ferroelectric material with pyroelectric, piezoelectric, electro-optic and photoelastic properties, and a high Curie point (~1200 °C) [13]. Ferroelectric materials have applications in constructing various tools such as memory devices, actuators, surface acoustic wave filters, heat sensors, and light wavelength converters, owing to their characteristic properties. Therefore, it is important to understand the mechanical, electrical, and optical characteristics of ferroelectric materials [4]. These characteristics are strongly related to the structure of lattice defects in these materials.

The lattice defects of LiNbO3 have been studied for several decades; the point defects [510] have received attention because of their non-stoichiometry due to complex oxides. On the other hand, it seems that the plane and line defects of LiNbO3 receive less attention. It has been reported that twinning deformation occurs on \( \{ {12\overline{1} 0} \}/ {\langle}\overline{1} 011{\rangle} \) [1113] and that the basal glide system [0001] \( {\langle}\overline{1} 2\overline{1} 0{\rangle} \) seems easier to activate than the prismatic glide system \( \{ {\overline{1} 2\overline{1} 0} \}/ {\langle}\overline{1} 101{\rangle} \) at high temperature [14]. The microscopic structures of the twins and dislocations in LiNbO3 have not been understood well because few studies have been conducted using transmission electron microscopy (TEM).

In the present study, a LiNbO3 bicrystal with a \( \{ {\overline{1} 2\overline{1} 0} \}/ {\langle}10\overline{1} 0{\rangle} \) low-angle tilt grain boundary was fabricated by diffusion bonding of two single-crystals, to investigate the boundary structure and characterize the microscopic structure of a dislocation with \( b = { 1}/ 3{\langle}\overline{1} 2\overline{1} 0{\rangle} \). Then the resultant boundary was investigated using high-resolution TEM (HRTEM) at room temperature, and its structure was examined from a crystallographic viewpoint. Here, the bicrystal experiment using a low-angle grain boundary is powerful for characterizing the dislocation structure because an ideal array of dislocations is formed on the boundary [1518].

A dislocation with \( b = { 1}/ 3{\langle}\overline{1} 2\overline{1} 0{\rangle} \) in LiNbO3 has a minimum translation vector on the (0001) basal plane and corresponds to the basal dislocation that brings about the basal glide system \( ( {000 1} ) {\langle}\overline{1} 2\overline{1} 0{\rangle} \). It is well known that a basal dislocation with \( b = { 1}/ 3{\langle}\overline{1} 2\overline{1} 0{\rangle} \) in α-Al2O3 can dissociate into two partial dislocations with the Burgers vectors of \( 1/ 3{\langle}0 1\overline{1} 0{\rangle} \) and \( 1/ 3{\langle}\overline{1} 100{\rangle} \) [19, 20], where α-Al2O3 has a rhombohedral \( R\overline{3} c \) structure, which is similar to the R3c structure of LiNbO3 at room temperature. Thus, it is worth paying attention to the structure of a dislocation with \( b = 1/ 3{\langle}\overline{1} 2\overline{1} 0{\rangle} \) in LiNbO3. It should be noted that the core structure of the boundary dislocations might differ from that of the lattice dislocations owing to the proximity between the neighboring boundary dislocations and the diffusion bonding process in a bicrystal.

Crystal structure of LiNbO3

LiNbO3 has a rhombohedral R3c structure at room temperature and \( R\overline{3} c \) above its Curie point [1, 2]. Figure 1 shows a schematic illustration of the crystal structure in the ferroelectric state at room temperature. Figure 1a–c show the configuration of constituent ions along \( [ {1\overline{2} 10} ],[ {10\overline{1} 0} ] \), and [0001], respectively. Here oxygen ions were approximated to be in a hexagonal close packed (hcp) arrangement for the ease of understanding the crystal structure. Note that the actual sites of the oxygen ions are displaced slightly on (0001) from the regular hcp arrangement. The crystal structure can be thought of as having a hcp arrangement of oxygen ions, where 2/3 of the octahedral sites of oxygen ions are occupied by cations. As seen in Fig. 1, LiNbO3 has a complicated crystal structure, which may be an obstacle in understanding the structure of plane and line defects.
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Fig. 1

Schematic illustration showing the crystal structure of LiNbO3 at room temperature. The arrangement of ions along \( [ {1\overline{2} 10} ],\,[ {10\overline{1} 0} ] \), and [0001] are represented in (ac), respectively. Two kinds of stacking of O ions along [0001] are denoted by large open circles and large half-shaded circles. The vectors of \( 1/ 3[ {\overline{1} 2\overline{1} 0} ],\,[ {10\overline{1} 0} ] \), and [0001] correspond to the minimum translation vectors along their directions of LiNbO3

Experimental procedure

A congruent LiNbO3 single-crystal grown by the Czochralski method [21, 22] was used to fabricate a bicrystal with a \( \{ {\overline{1} 2\overline{1} 0} \}/ {\langle}10\overline{1} 0{\rangle} \) low-angle tilt grain boundary. Figure 2 shows a schematic illustration of the two pieces of single-crystal plates before bonding and the fabricated bicrystal after bonding. The size of each single-crystal plate was set as 10 × 10 × 1 mm3 to prepare a bicrystal with a size of 10 × 10 × 2 mm3. The surfaces of the single-crystal plates were successively polished using diamond suspensions to achieve a mirror finish. The two single-crystal plates were subsequently joined by diffusion bonding in air at 800 °C for 10 h under an additional pressure of 0.1 MPa. As shown in the figure, the \( ( {\overline{1} 2\overline{1} 0} ) \) single-crystal plate was bonded with the plate 1° off from the exact \( ( {\overline{1} 2\overline{1} 0} ) \) plane about the \( [ { 10\overline{1} 0} ] \) axis, resulting in the fabricated bicrystal with a 1° asymmetric \( \{ {\overline{1} 2\overline{1} 0} \}/ {\langle}10\overline{1} 0{\rangle} \) tilt grain boundary. In this case, edge-type dislocations with b = 1/3\({\langle} \overline{1} 2\overline{1} 0{\rangle} \) will primarily form on the boundary [16, 23].
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Fig. 2

Schematic illustration showing the shapes and crystallographic orientation relationship of the bicrystal before and after diffusion bonding. a Two pieces of single-crystal plates for bonding. b Fabricated bicrystal after bonding. The edge-type of a perfect dislocation array is illustrated here, which can be formed ideally

The directions of polarization in the two plates were set up to be the same in the bonding process. LiNbO3 has polarization only along <0001> because it has the R3c structure and the structure or character along the [0001] direction differs from that along the opposite direction of \( [ { 000\overline{1} } ] \). Therefore, the temperature of 800 °C for bonding in this study is selected to be sufficiently lower than the Curie point of ~1200 °C. Here, 800 °C (1073 K) corresponds to about 0.7 times the melting point of 1253 °C (1526 K). The polarization in the two plates will be conserved during the bonding process.

Figure 3 shows optical photographs of the bicrystal before and after diffusion bonding. The two plates exhibit interference fringe on the interface, as seen in (a), because the two plates have not yet been joined and have space between them, while the fabricated bicrystal in (b) exhibits little fringe and looks like a single-crystal. This indicates that the two single-crystal plates were successfully joined by diffusion bonding. The grain boundary of this fabricated bicrystal was observed by TEM. The specimens for the observation were prepared using a standard technique involving mechanical grinding to a thickness of 0.1 mm, attaching with a stainless-steel single-hole mesh for reinforcement, dimpling to a thickness of about 30 μm, and ion beam milling to an electron transparency at about 4 kV. TEM observation was conducted using a conventional TEM (c-TEM; JEOL JEM-2100, 200 kV) and a high-resolution TEM (HRTEM; JEOL JEM-4010, 400 kV).
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Fig. 3

Optical photographs of the bicrystal before and after diffusion bonding. a Double single-crystal plates display interference fringe on the interface since the two plates are not joined. b The fabricated bicrystal displays little fringe

Results

Figure 4a shows a typical bright-field TEM image of the grain boundary taken along \( [ { 10\overline{1} 0} ] \). In the case of a bright-field image, as seen in the figure, the particle-like regions of 100–200 nm with moiré are distinctly observed at the boundary, in addition to the boundary dislocations that compensate the misorientation angle on this boundary. Figure 4b shows a selected-area electron diffraction pattern taken from a large region with a diameter of about 900 nm, which includes the particle-like regions. The particle-like regions with moiré have the same crystal structure as the bulk because the diffraction patterns are the same. The regions with moiré are considered to have little distortion of the crystal lattice. This indicates that LiNbO3 may allow a slight change in the lattice parameter locally, as expected from its application as a piezoelectric material. Further discussions about the particle-like regions with moiré are presented in the Appendix 1 so that we can focus on the structure of boundary dislocations in the main text.
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Fig. 4

Results of TEM observation of the grain boundary along \( [ {10\overline{1} 0} ] \). a Bright-field image under a two-beam condition. b Selected-area electron diffraction pattern taken from a large area, including the particle-like regions in (a). I and II are magnified patterns to show detail. c HRTEM image containing four boundary dislocations in the region with less moiré

Figure 4c shows a TEM image of the boundary dislocation array taken along \( [ { 10\overline{1} 0} ] \) from the region with less moiré. This image is taken using a HRTEM technique, although the magnification is not very high. The white arrows in the figure indicate the position of the boundary dislocations. The periodic spacing between two neighbor dislocations is about 27 nm. Here the relation between the spacing d and the misorientation angle in a low-angle tilt grain boundary, θ, is given by θ = b/d, where b is the size of the edge component of the Burgers vector of the boundary dislocations, according to the Frank formula [24]. In this boundary, the minimum translation vector normal to the (\( \overline{1} 2\overline{1} 0 \)) boundary plane is \( 1/ 3[ {\overline{1} 2\overline{1} 0} ] \), the size of which corresponds to a representative lattice constant of LiNbO3, that is, the Burgers vector of the boundary dislocations should be \( 1/ 3[ {\overline{1} 2\overline{1} 0} ] \). The calculated misorientation angle is 1.1°.

Figure 5 shows a typical HRTEM image of a boundary dislocation taken along \( [ {10\overline{1} 0} ] \), which directly includes one of the boundary dislocations. It should be noted that two neighbor lattice discontinuities clearly appear in the magnified HRTEM image, as indicated by the arrows. This implies that a boundary dislocation dissociates into two partial dislocations with a narrow separation distance. Here, the periodicity of the bright points in the HRTEM image corresponds to that of 1/3[0001] along [0001] and \( 1/ 6[ {\overline{1} 2\overline{1} 0} ] \) along \( [ {\overline{1} 2\overline{1} 0} ] \). See Appendix 2 for details on the bright points on the HRTEM image of LiNbO3.
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Fig. 5

Magnified HRTEM image including a boundary dislocation along \( [ {10\overline{1} 0} ] \)

Figure 6a shows the magnified HRTEM image from Fig. 5 with the Burgers circuits around the two lattice discontinuities. It can be clearly seen that the two lattice discontinuities are the two dislocations with an edge component of \( 1/ 6[ {\overline{1} 2\overline{1} 0} ] \). Figure 6b shows an inverse Fast-fourier-transferred (FFT) image reconstructed from a mask-applied FFT image of the area shown in Fig. 6a. It is found in Fig. 6b that the separation distance between the two dislocations was 2.7 nm along [0001], while the two are adjacently located along \( [ {\overline{1} 2\overline{1} 0} ] \). It can be said from direct observation by HRTEM that a boundary dislocation of \( b = 1/ 3[ {\overline{1} 2\overline{1} 0} ] \) dissociates into two partial dislocations with an edge component of \( 1/ 6[ {\overline{1} 2\overline{1} 0} ] \) by the narrow separation distance along [0001]. It should be noted that the separation distance of 2.7 nm appears to slightly vary from one dislocation to another one.
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Fig. 6

a Magnified HRTEM image of Fig. 5 with the Burgers circuits around the two lattice discontinuities. b Inverse FFT image reconstructed from a mask-applied FFT image of (a)

Discussion

Structure of a dislocation of \( b = 1/3[ {\overline{1} 2\overline{1} 0} ] \)

It is found experimentally that a dislocation of \( b = { 1}/ 3[ {\overline{1} 2\overline{1} 0} ] \) in LiNbO3 dissociates into two partial dislocations along [0001]. If a dislocation dissociates, a stacking fault with the fault vector that corresponds to the Burgers vector of a partial dislocation is formed between the partial dislocations. Figure 7a, b show schematic illustrations of a dissociated dislocation and a dissociated dislocation’s array at the boundary, respectively. The stacking sequence of the (\( \overline{1} 2\overline{1} 0 \)) plane along [\( \overline{1} 2\overline{1} 0 \)] in LiNbO3 is …αβαβ…, as shown in (a). Here, we note that the stacking fault plane of (\( \overline{1} 2\overline{1} 0 \)) is not on the (0001) basal plane. This means that a dislocation dissociates by climb and not by glide. Although climb requires point defect diffusion, this is easily possible during diffusion bonding at elevated temperatures. According to elasticity theory, the climb-dissociated configuration is more stable than the glide-dissociated one. Figure 7b explains the periodic formation of partial dislocations caused by the dissociation of dislocations and stacking faults between the partials. Here we discuss the Burgers vectors of separated partial dislocations and the structure of a stacking fault formed on (\( \overline{1} 2\overline{1} 0 \)) between the partials.
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Fig. 7

Schematic illustration showing the structure of observed dissociated dislocations. a Singular dissociated dislocation, which corresponds to the observed dislocation in Fig. 5. The stacking sequence of the \( ( {\overline{1} 2\overline{1} 0} ) \) plane along \( [ {\overline{1} 2\overline{1} 0} ] \) is represented by α and β. A stacking fault is formed along the \( ( {\overline{1} 2\overline{1} 0} ) \) plane between β2 and β3. b Plural dissociated dislocation. Partial dislocations and stacking faults are periodically formed on the boundary

Figures 8 and 9 show schematic illustrations of the cation and anion sublattices viewed along [0001]. The stacking sequence along [\( \overline{1} 2\overline{1} 0 \)] is also represented by α and β in these illustrations. For both sublattices, a perfect stacking sequence is shown in (a), while the stacking sequence, including the stacking faults on the (\( \overline{1} 2\overline{1} 0 \)) plane with shears of \( 1/ 2[ { 10\overline{1} 0} ],\,1/3[ {10\overline{1} 0} ] \), and \( 1/3[ {\overline{1} 010} ] \), are in (b–d), respectively. As shown in Fig. 6, a partial dislocation has an edge component of \( 1/ 6[ {\overline{1} 2\overline{1} 0} ] \). If it has only the edge component, that is, if the Burgers vector is \( 1/ 6[ {\overline{1} 2\overline{1} 0} ] \), the structure of the stacking fault between two partial dislocations corresponds to the illustrations in Figs. 8b, 9b. However, the structure in these figures gives rise to the stacking fault of both the cation and anion sublattices of LiNbO3, since the \( 1/ 2[ 10\overline{1} 0] \) shear does not coincide with a translation vector of oxygen ions of \( 1/ 3{\langle} 10\overline{1} 0 {\rangle} \) or \( 1/ 3{\langle} 01\overline{1} 0 {\rangle} \) or \( 1/ 3{\langle} \overline{1} 100 {\rangle} \) on the [0001] plane. In contrast, the structures in Fig. 8c, d show the stacking faults only for the cation sublattices because their shear vectors correspond to the translation vector of oxygen ions as shown in Fig. 9c, d. In this case, the dislocation of \( b_{1} = { 1}/ 3[0 1\overline{ 1} 0] \) or \( b_{{\mathbf{1}}} = 1/3[\overline{1} 100] \) has both the edge component of \( 1/ 6[ {\overline{1} 2\overline{1} 0} ] \) and the screw component of \( \pm 1/ 6[ { 10\overline{1} 0} ] \), and is therefore possible to be a partial dislocation. The vectors of b1 and b2 are also shown in Fig. 8a. Thus, the three types of stacking faults in (b–d) of Figs. 8, 9 are possible as the stacking fault between the partial dislocations.
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Fig. 8

Schematic illustration showing the structure of the cation sublattice viewed along [0001]. a Perfect stacking sequence. Stacking sequences in bd, show the cation arrangement, including stacking faults on the \( ( {\overline{1} 2\overline{1} 0} ) \) plane with the shears of \( 1/ 2[ {10\overline{1} 0} ],\,1/3\,[ {10\overline{1} 0} ] \), and \( 1/ 3[ {\overline{1} 010} ] \), respectively. The vectors of b1 and b2 are the Burgers vectors of two partial dislocations, which are supposed to be formed by the dissociation

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Fig. 9

Schematic illustration showing the structure of the anion sublattice viewed along [0001]. a Perfect stacking sequence. Stacking sequences in bd, show the anion arrangement, including stacking faults on the \( ( {\overline{1} 2\overline{1} 0} ) \) plane with the shears of \( 1/ 2[ {10\overline{1} 0} ],\,1/3\,[ {10\overline{1} 0} ] \), and \( 1/ 3[ {\overline{1} 010} ] \), respectively. It should be noted that the stacking sequences in (c) and (d) go back to a perfect stacking sequence

It is well known that a basal dislocation of \( b = { 1}/ 3{\langle} \overline{1} 2\overline{1} 0 {\rangle} \) in α-Al2O3 often dissociates into two partial dislocations with the Burgers vectors of \( 1/ 3{\langle} 0 1\overline{1} 0 {\rangle} \) and \( 1/ 3{\langle}\overline{1} 1 00 {\rangle} \) [19, 20]. The partial dislocation in α-Al2O3 has a screw component of \( \pm 1/ 6{\langle}10\overline{1} 0 {\rangle} \) in addition to an edge component of \( 1/ 6{\langle} \overline{1} 2\overline{1} 0 {\rangle} \) because a stacking fault in both the cation and anion sublattices has much higher energy than that in only the cation sublattice. This can be applied to LiNbO3 with a similar crystal structure. It is suggested that the dislocation of \( b = 1/ 3[ {\overline{1} 2\overline{1} 0} ] \)] in LiNbO3 should dissociate into the two partial dislocations of \( b_{{\mathbf{1}}} = { 1}/ 3[0 1\overline{ 1} 0] \) and \( b_{2} = 1/3[\overline{1} 100] \).

As for the two types of stacking faults in (c) and (d) in Fig. 8, the structure along \( [ {\overline{1} 2\overline{1} 0} ] \) in (c) corresponds to that along \( [ {1\overline{2} 10} ] \) in (d) as can be seen from β2 and β3. Thus, we note that the structures of the two stacking faults have a mirror symmetry relationship. The energy of these two stacking faults should be the same. On the other hand, the atomic arrangement around the core of a partial dislocation may depend on the location of an extra half plane of the core to the polarization direction along [0001]. That is, the atomic structure at the core of a partial dislocation may be dependent on the polarization direction. The influence of polarization on the core structure, which is a specific issue for ferroelectric materials, will be a subject for future studies.

Stacking fault energy and separation distance

A dislocation with \( b = { 1}/ 3[ {\overline{1} 2\overline{1} 0} ] \) at the boundary dissociated into two partial dislocations with a separation distance of 2.7 nm, forming a \( ( {\overline{1} 2\overline{1} 0} ) \) stacking fault. The separation distance is determined by a balance of two forces, the repulsive elastic force between partial dislocations and the attractive force due to the stacking fault energy, acting in the dissociated dislocation. According to the Peach–Koehler’s equation [25], the balance in the present dislocation can be expressed by the following equation:
$$ \gamma = \frac{{\mu b_{\text{p}}^{2} (2 + \nu )}}{8\pi r(1 - \nu )}, $$
(1)
where γ is the stacking fault energy, μ is the shear modulus (66 GPa), ν is the Poisson’s ratio (0.25), bp is the size of the Burgers vectors of the partial dislocations with \( b_{1} = { 1}/ 3[0 1\overline{ 1} 0] \) and \( b_{2} = 1/3[\overline{1} 100] \), and r is the separation distance. It should be taken into account that the separation distance is affected by all the elastic repulsive forces, which act on a partial dislocation from all other dislocations at the boundary [15, 17, 23]. The balance between repulsive and attractive forces in this boundary is given as
$$ \gamma = \frac{{\mu b_{\text{p}}^{2} (2 + \nu )}}{8\pi (1 - \nu )d}\sum\limits_{n = 0}^{\infty } {\left( {\frac{1}{n + \alpha } - \frac{1}{n + 1 - \alpha }} \right)} , $$
(2)
where d is the periodic spacing between neighbor boundary dislocations and α is the ratio of r to d. Here, α = 0.10 is obtained owing to d = 27 nm and r = 2.7 nm. By substituting the values of d and α in Eq. 2, the energy of a (\( 1\overline{2} 10 \)) stacking fault is estimated to be about 0.25 J/m2. The elastic constants used for the equations were derived from the former studies [3, 26, 27]. The equations are based on a conventional elastic theory for an isotropic elastic medium. Accordingly, the evaluation using Eq. 2 may slightly lose accuracy because of the specific crystal structure of LiNbO3 with polarization. Here, the c11, c12, and c44 are reported to be 2.03 × 1011, 0.573 × 1011, and 0.595 × 1011 (N/m2), respectively [26]. The ratio of c44 to (c11 − c12)/2 is calculated to be 0.82. This value indicates a degree of elastic anisotropy, which does not seem large. Therefore, we could apply isotropic elasticity theory. Note that, because of the extremely narrow separation distance, the stacking fault between the partials may locally have a different atomic arrangement from the ideal stacking fault. This would lead to a slight error in the stacking fault energy derived from the dissociated dislocation.

Conclusion

A LiNbO3 bicrystal with a \( \{ {\overline{1} 2\overline{1} 0} \}/ {\langle} 10\overline{1} 0{\rangle}\) 1° low-angle tilt grain boundary was fabricated by the diffusion bonding of two single crystals with a controlling crystallographic orientation relationship and polarization, to investigate the structure of the resultant grain boundary and boundary dislocations by TEM. The particle-like regions of 100–200 nm with moiré are distinctly observed at the fabricated boundary, in addition to the boundary dislocations that compensate the misorientation angle at the boundary. HRTEM observation successfully found that the dislocation with \( b = { 1}/ 3{\langle} \overline{1} 2\overline{1} 0{\rangle} \) dissociates into two partial dislocations along < 0001 > and with an adjacent location along \( {\langle} \overline{1} 2\overline{1} 0{\rangle} \). This indicates that the dislocations dissociate by climb, not by glide. It is suggested that the Burgers vectors of the partial dislocations should be \( 1/3 {\langle} 01\overline{1} 0 {\rangle} \) and \( 1/3{\langle} \overline{1} 100 {\rangle} \). The stacking fault formed on \( \{ {\overline{1} 2\overline{1} 0} \} \) by the dissociation is thought to have either of the two types of structures with mirror symmetry, which should be the same in energy. By applying the separation distance in a partial dislocation pair of 2.7 nm to the equation based on isotropic elastic theory, the stacking fault energy on \( \{ {\overline{1} 2\overline{1} 0} \} \) is estimated to be about 0.25 J/m2.

Acknowledgements

The authors wish to express their gratitude to Prof. Y. Ikuhara and Prof. N. Shibata for fruitful discussion and encouragement. They also thank Ms. N. Uchida for technical support in TEM operation. Part of this work was supported by a Grant-in-Aid on Priority Areas “Nano Materials Science for Atomic-scale Modification” (no. 19053001). The authors acknowledge the support of National Center for Electron Microscopy, Lawrence Berkeley Laboratory on the simulated HRTEM images, which is supported by the U.S. Department of Energy under Contract # DE-AC02-05CH11231. E. T. was supported by the JSPS postdoctoral fellowship for research abroad.

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© Springer Science+Business Media, LLC 2012