α-Al2O3 interface by geometric phase analysis and dislocation density tensor analysis

Materials Characterization 106 (2015) 308–316

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Investigation of misfit dislocations in α-Fe2O3/α-Al2O3 interface by geometric phase analysis and dislocation density tensor analysis Y. Wang ⁎, X.P. Liu, G.W. Qin Laboratory for Anisotropy and Texture of Materials (Ministry of Education), Northeastern University, Shenyang 110004, China

a r t i c l e

i n f o

Article history: Received 26 March 2015 Received in revised form 16 June 2015 Accepted 16 June 2015 Available online 18 June 2015 Keywords: Sapphire Misfit dislocation Transmission electron microscopy Geometric phase analysis Dislocation density tensor analysis

a b s t r a c t The misfit dislocations in α-Fe2O3/α-Al2O3 heterostructure interfaces were investigated by high-resolution transmission electron microscopy (HRTEM), geometric phase analysis (GPA) and dislocation density tensor analysis. The misfit dislocations form a two-dimensional network structure at the interface. We characterized the atomic configurations and strain distribution of misfit dislocations by HRTEM and GPA. The observation indicates that one and two extra half planes/strain fields exist in the dislocation cores imaged along the [1100] and [1120] direction, respectively. Dislocation density tensor analysis gave a very high accuracy in determining the Burgers vectors and proved useful in accurately localizing the dislocation distribution in the core region. The results show that all the dislocations have the same Burgers vector, however the different space distributions of dislocation density, which may be attributed to the differences of atomic configuration in dislocation cores. Classical elasticity theory was found to be in agreement with the 3D visualizations of dislocation density tensor. The relationships between atomic configuration, dislocation density distributions, and strain distributions in dislocation cores were investigated, which can be described briefly as: one/two extra half planes induce one/two Burgers vectors in dislocation density distributions; one/two Burgers vectors induce one/two strain fields around dislocation core. Finally by comparing the experimental strain distributions with dislocation models (Peierls–Nabarro model and Foreman model), we found that all the dislocations follow the Foreman model (a = 2), which indicates that they have the same spatial extension of strain field. This work demonstrates the superiority of dislocation density tensor analysis in the investigation of misfit dislocations, particularly the dislocations with complicated core structure. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The properties of a thin film are influenced by many factors that include stoichiometry, microstructural homogeneity, defect concentrations and impurity levels [1–4]. It is generally accepted that defects in a thin film can be initiated from the interface between the film and the substrate. Thus, it is necessary to study the detailed defect characteristics at the interface. The most common defects at the interface are misfit dislocations, which can relax the mismatch between the film and substrate. Consequently detailed analysis of misfit dislocations has been at the focus of materials research for a long time [5–7]. In recent years, a new method based on the combination of highresolution transmission electron microscopy (HRTEM), geometric phase analysis (GPA) and dislocation density tensor analysis has been developed to investigate partial dislocations [8] and misfit dislocations [9–12]. The atomic configuration, strain mapping, dislocation density distribution and Burgers vectors of dislocations can be obtained simultaneously by this method. Such a detailed investigation of dislocation core structure is quite important as it influences many aspects of ⁎ Corresponding author.

http://dx.doi.org/10.1016/j.matchar.2015.06.012 1044-5803/© 2015 Elsevier Inc. All rights reserved.

material behavior, including mechanical, optical and electromagnetic properties. This method, proposed by Kret et al. during the investigation of misfit dislocations in GaAs/CdTe interface [9], starts with the GPA for extracting the lattice distortion field near dislocation cores from HRTEM. Next, the dislocation density distribution is calculated from the lattice distortion field. Lastly, Burgers vectors are precisely determined by mathematical integration of the dislocation density distribution over dislocation core regions. The results show that all the misfit dislocations are Lomer dislocations with two dislocation density peaks corresponding to in-plane components of two 60° dislocations [9]. This agrees with the fact that a Lomer dislocation can be interpreted as two merged 60° dislocations coming from different {111} planes. This method was also employed to localize and measure the dislocation core of partial dislocations in wurtzite GaN by Kioseoglou et al. [8]. By comparing experimental results to atomistic simulations, this method was proved useful in accurately localizing core region and obtaining core radius. The experimental core radius was in good agreement with second-order gradient elasticity theory. More recently, Wang et al. applied this method to analyze the mechanism of formation of misfit dislocations in GaSb/GaAs interfaces and the source of threading dislocation in GaSb epitaxial films [10–12]. By the dislocation density tensor analysis, it

Y. Wang et al. / Materials Characterization 106 (2015) 308–316

is shown that there are three types of misfit dislocations in GaSb/GaAs interfaces: shuffle and glide set Lomer dislocations and 60° dislocation pairs, and the dominant mechanism underlying the formation of misfit dislocations is the glide and reaction of 60° dislocations [10]. The 60°, Lomer and 60° pair misfit dislocations are proved to be the source of three threading dislocations in GaSb epitaxial films: mixed, edge and pair of mixed types respectively [11]. In addition, GPA was used to map the strain field distribution around 60°, Lomer and 60° pair misfit dislocations. A detailed comparison of experimental strain mapping to the available dislocation models (Peierls–Nabarro and Foreman model) applied in bulk materials proves that these models can also be adapted to the misfit dislocations at lattice mismatched interface, where it is shown that the strain field of the 60° dislocation follows the Foreman model (a = 1.8), in case of the Lomer and 60° dislocation pair, the Foreman model (a = 2.5) and Peierls–Nabarro model apply for εxx and εyy, respectively [12]. As mentioned above, all the research works in the past few years have been focused on dislocations of cubic semiconductor materials, such as GaSb and GaAs. In this work, a method, based on the combination of HRTEM, GPA and dislocation density tensor analysis, was employed to investigate misfit dislocations in rhombic dielectric materials α-Fe2O3/α-Al2O3 interface, since the epitaxy of iron oxide thin films is of great interest for application in heterogeneous catalysis, magnetic thin film devices, surface geochemistry, and integrated microwave devices [13–16]. The atomic configuration of misfit dislocations imaged along the [1100] and [1120] direction was investigated by HRTEM. The strain fields around the dislocation core were mapped by GPA and compared with theoretical models. From the lattice distortion field, the dislocation density distribution was calculated. The Burgers vectors, space distributions of dislocation density and 3D visualizations of the dislocation density tensor in dislocation core were measured and compared with theoretical value of Burgers vectors, atomic configuration in dislocation cores and elasticity theory models, respectively. This work demonstrates the superiority of dislocation density tensor analysis in the investigation of misfit dislocations, particularly the dislocations with complicated core structure.

the geometric phases of any two lattice fringes suffice, providing their reciprocal lattice vectors are non-collinear. In practice, the lattice fringes giving the best signal-to-noise are chosen. The two-dimensional deformation tensor can be obtained by differentiating the displacement field, defined as follows [17]:  ε¼

εxx εyx

1 ∂ux ∂y C C: ∂uy A

∂ux B ∂x ¼B @ ∂uy

The method of GPA, developed by Hÿtch et al. [17], provides a powerful tool for measuring strains quantitatively as revealed in highresolution transmission electron microscopy (HRTEM). A precision of 3 pm was achieved during the study of strain field around dislocations in silicon by this technique [18]. An HRTEM image formed at a zone axis of a crystal can be considered as a set of interference fringes corresponding to the atomic planes of the specimen. GPA analyzes these interference fringes individually to extract the information concerning strain. In particular, the technique measures the displacement of lattice fringes with respect to a perfect lattice (for example provided by a region of unstrained substrate). The method is based on calculating the local Fourier components of lattice fringes. The phase of these local Fourier components, or geometric phase Pg(r), is directly related to the component of the displacement field, u(r), in the direction of the reciprocal lattice vector, g [17]: ð1Þ

Given the geometric phase of two diffracted beams, the displacement field can be determined [17]: ð2Þ

where a1 and a2 are the real-space basis vectors corresponding to the reciprocal lattice defined by g1 and g2 (i.e. ai · gj = δij). In principle

ð3Þ

∂y

2.2. Dislocation density tensor analysis The dislocation density tensor characterizes the strength of infinitesimal dislocation at each point in a continuously dislocated crystal and provides a measurement of the Burgers vector [8–10]. The local Burgers vector is given by the line integral (closed curve around the dislocation) of the lattice distortion tensor [19] as well as the surface integral of the dislocation density tensor [20]: b ¼ ∬ α  dS ¼ −∮ ε  dl S

ð4Þ

l

where surface S is bounded by a closed curve l, α and ε are the dislocation density tensor and the lattice distortion tensor, respectively. In 2D, the components of the dislocation density tensor αx and αy are extracted from the lattice distortion tensor as follows [8]: 2

αx ¼

2

∂εxx ∂εxy ∂ ux ∂ ux − ¼ − ∂y ∂x ∂x∂y ∂y∂x 2

2.1. Geometric phase analysis (GPA)

 1
P g1 ðrÞa1 þ P g2 ðrÞa2 2π

0

Thus the biaxial strains εxx and εyy are derived to illustrate the local lattice displacement from the reference lattice. In this work, GPA was performed in Digital Micrograph using Koch's FRWR tools plugin, which is based on the methods of Hÿtch et al. [17].

αy ¼ −

uðrÞ ¼ −

εxy εyy



∂x

2. Materials and methods

P g ðrÞ ¼ −2πg  uðrÞ:

309

ð5Þ 2

∂ε yy ∂εyx ∂ uy ∂ uy þ ¼− þ : ∂x ∂y ∂y∂x ∂x∂y

ð6Þ

Experimentally, the lattice distortion tensor can be obtained by GPA. In fact, for sufficient smooth displacement field (ux, uy), the dislocation density tensor (αx, αy) vanishes identically since the mixed derivatives in Eqs. (5) and (6) are equal. However, the continuity conditions break down at the dislocation core. As a result, the two in-plane components of the tensor field take zero values over the whole region except at the dislocation core position. Integrating the in-plane component over the dislocation core region we obtain the corresponding Burgers vector component [8]. 2.3. Material synthesis and structural characterization High-purity α-Al2O3 single crystals (sapphire) having the c-axis normal to the surface were purchased from Valley Design Corporation. The wafers were polished to a mirror-like finish and were annealed at 1500 °C for 6 h in air to remove mechanical damage before ion implantation. Fe ions were implanted into (0001)-oriented sapphire wafers at energy of 50 keV to a dose of 1 × 1017 ions/cm2 at room temperature. The wafers were tilted 7° with respect to the ion beam to avoid channeling effects. The Fe-implanted samples were subsequently annealed at 1000 °C for 2 h in an oxidizing (air) atmosphere. Specimens for TEM were thinned mechanically, and then thinned by ion milling. The microscopy observation was carried out using a Tecnai G2 F30 field emission transmission electron microscope with spherical aberration Cs 1.2 mm and TEM point resolution 0.2 nm. The microscope is equipped with a bottom-mounted 2048 × 2048 pixel slow-scan CCD

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camera. Fast Fourier transformation (FFT) was carried out using a Digital Micrograph software package. 3. Results and discussion 3.1. Misfit dislocations imaged along the [1100] direction Cross-sectional high-resolution transmission electron microscopy (HRTEM) observations were performed in order to investigate the α-Fe2O3/α-Al2O3 interface, as shown in Fig. 1(a). The cross-sectioned plane is parallel to (1100) of sapphire as indicated by the inserted Fourier transform of Fig. 1(a). The lower region is sapphire, and the upper region is α-Fe2O3. It can be seen from the image that the interface is sharp, flat and free of second phases or intermediate layers. To distinguish the misfit dislocations at the interfaces clearly, the HRTEM image was Fourier filtered by choosing the (1120) diffraction spots of α-Fe2O3 and α-Al2O3 only, and the result is shown in Fig. 1(b). An approximately periodic array of misfit dislocations is found along the interface. The spacing between misfit dislocations has an average value of 4.1 nm, which corresponds to approximately 16 α-Fe2O3 (1120) planes and 17 α-Al2O3 (1120) planes. Thus it can be concluded that the α-Fe2O3/ α-Al2O3 interfaces are semicoherent, that is coherent regions separated by misfit dislocations at the interfaces. Both the α-Fe2O3 and α-Al2O3 have the same corundum structure with R3c space group and close lattice parameters (a/c = 0.476 nm/1.299 nm and 0.504 nm/1.374 nm of α-Al2O3 and α-Fe2O3, respectively). The misfit dislocations can relax the lattice mismatch between α-Fe2O3 and α-Al2O3. Inserted in Fig. 1(b) is a magnified HRTEM image of one of the misfit dislocations. The image has been treated using the Fourier filter method to reduce the noise. The misfit dislocation core is characterized by an extra (1120) plane on the α-Al2O3 side of the interface and this extra half plane terminates at the interface. By drawing a Burgers circuit surrounding the misfit dislocation core, the projected Burgers vector (edge component) is determined to be a/6[1120]sapphire (~ 0.24 nm), where a is the lattice constant of α-Al2O3. This projected Burgers vector is parallel to the interface and accommodate the mismatch between the lattices of α-Fe2O3 and α-Al2O3 along the [1120] direction. The strain distribution around the dislocation core was investigated by geometric phase analysis (GPA). As shown in Fig. 1(a), the (0003) and (1120) spots were chosen for GPA and masks were placed around these spots to isolate them. The area between two adjacent dislocations was taken as the reference during GPA, since Wang et al. have found that a reference taken between two adjacent dislocations gives optimal

results for the determination of the contribution of the strain due to the lattice mismatch [12]. Taking the x-axis parallel to [1120] (along the interface) and the y-axis parallel to [0001], the strain distribution εxx, εxy, εyx and εyy can be calculated using GPA, as shown in Fig. 2(a)–(d). It is obvious that there are no y-component strain fields (εyx and εyy) in the dislocation core, from which it can be concluded that the ycomponent displacement field uy is zero according to Eq. (3). This can be attributed to the (1120) extra half plane perpendicular to the interface, which can induce x-component displacement field ux only. In the strain field along the x direction Fig. 2(a), there is a convergence region of strain around the edge dislocation core. On the α-Al2O3 side, in which there exist the extra half planes, the strains are negative and compressive, and on the other side, the strains are positive and tensile. The largest strain values occur in the immediate core region. The strain is smaller farther from the dislocation core. The color scale indicates strain changes of −15% to +15%. The dislocation density tensor was calculated from the strain field according to Eqs. (5)–(6). Fig. 2(e) shows the x-component of the dislocation density tensor αx, and the y-component of the dislocation density tensor αy is zero (not shown here). The dark dots stand for the dislocation density distribution. Integrating the αx over the dislocation core region we obtain the projected Burgers vector (0.242 nm), which is in agreement with the value obtained by Burgers circuit. It can be seen that all the dislocations have the same Burgers vector. A magnified αx of one dislocation and the corresponding 3D visualization are shown in Fig. 2(f) and (g). It is observed that the dislocation density distribution is confined to one dot (a pixel with 0.043 nm × 0.043 nm) and the whole region around this dot take almost zero values (less than 1 × 10−6). The shape of the dislocation density tensor component is a spike with its height corresponding to the projected Burgers vector. Two theories have been developed to describe the dislocation density tensor: classical elasticity theory [21] and gradient elasticity theory [22–25]. Following classical elasticity, the dislocation density tensor of an edge dislocation is given by [21]: α x ¼ bx δðxÞδðyÞ

ð7Þ

where δ is the Dirac delta function which has the value of infinity at (x, y) = 0 and the value zero elsewhere. The integral of the Dirac delta from any negative limit to any positive limit is equal to 1. The 3D visualization of the dislocation density tensor following de Wit's solution [21] yields a spike with its height equal to the Burgers vector. Gradient elasticity is an extension of classical elasticity based on the introduction of an internal length parameter (gradient coefficient) to account for atomistic effects beyond the ‘next neighbor atom’

Fig. 1. (a) High-resolution TEM image of the α-Fe2O3/α-Al2O3 interface viewed along [1100]sapphire, inserted by Fourier transform of image with the analyzed spot circled for GPA. (b) The Fourier filtered image of (a) by choosing the (1120) diffraction spots of α-Fe2O3 and α-Al2O3 only. The inset is the Fourier filtered HRTEM image of one of the misfit dislocations.

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Fig. 2. (a–d) The strain distribution εxx, εxy, εyx and εyy around the misfit dislocations obtained by GPA. All the figures have the same color scale as inserted in (a). (e) The x-component of the dislocation density tensor αx. (f, g) A magnified αx of one dislocation and the corresponding 3D visualization.

interactions. The gradient elasticity theory was applied to eliminate singularities from dislocation lines and obtain smoother distributions for the strain fields near and on the dislocation line [25]. It is obvious that the dislocation density tensor in this study, which yields a spike with its height corresponding to the projected Burgers vector, follows the classical elasticity theory. In the case of partial dislocations in wurtzite GaN, the shape of the dislocation density tensor components is conical with radius of 0.3 nm, in agreement with gradient elasticity theory [8]. Several theoretical models have been used to describe the strain fields of the edge dislocation in bulk materials. The Peierls–Nabarro (P–N) dislocation model is one of the most important models in edge dislocations developed by Peierls and Nabarro [26,27], and Foreman proposed an improved model based on the P–N model [28]. In the improved model, the P–N dislocation model is extended to a family of edge dislocations of greater widths by introducing the factor a. Actually the P–N model is equivalent to the Foreman model when a = 1. In the Foreman model, a is used to control the spatial extension of the strain field, which may be related to the elastic behavior of the investigated material. The validity of these models has been verified experimentally in some bulk materials, for instance, the Peierls–Nabarro model in gold [29] and the Foreman model in CeO2 [30], graphene [31] and germanium [32]. According to the P–N dislocation model, the strain of an edge dislocation along the x direction is given by [26,27]: εxx ¼

b ð1−νÞy π 4ð1−ν Þ2 x2 þ y2

ð8Þ

where x and y are the respective right-angle coordinates centered on the dislocation core position, b is Burgers vector, and ν is Poisson's ratio. According to Foreman dislocation model, the strain of an edge dislocation along the x direction is given by [28]: εxx ¼

  2 bð1−νÞ 4ð1−νÞ yx2 þ 2a3 −a2 y3 h i2 π 4ð1−νÞ2 x2 þ a2 y2

ð9Þ

where x and y are the respective right-angle coordinates centered on the dislocation core position, b is Burgers vector, ν is Poisson's ratio, and a is an alterable factor related to the width of the dislocation. In the case of a bulk material, the spatial behavior of the strain can be easily modeled [29–32], however, at the hetero-interface, the situation is more complicated, since the film's Poisson's ratio may be different from the substrate's. In this study, the strain fields of misfit dislocations were modeled using the Poisson's ratio of sapphire and α-Fe2O3 (hematite), separately. The calculated strain fields were compared with experimental results. Fig. 3(a)–(f) shows the εxx strain fields of the P–N model and Foreman models (from a = 2 to a = 6) calculated using the Poisson's ratio ν = 0.29 of sapphire [33], Burgers vectors b = 0.24 nm, and the origin situated at the dislocation core. For the comparison, a magnified strain field of one of the misfit dislocations in Fig. 2(a) is shown in Fig. 3(g). The color scale is the same for both experiment and simulation. It is obvious that the agreement of the experimental result and the Foreman model with a = 2 is best. Considering the different Poisson's ratios of sapphire and hematite, the εxx strain fields of the Foreman model with a = 2 were also calculated using the Poisson's ratio ν = 0.25 of α-Fe2O3 [34], as shown in Fig. 3(i). To analyze the degree of agreement between the experiment and theory in more detail, the comparison of the experimental results and theoretical models calculated using different Poisson's ratios was carried using circular profiles centered around the dislocation, as shown in Fig. 3(g), for two radii r = 0.3 nm and 0.6 nm. The result is shown in Fig. 3(j). It is shown that the strain fields calculated using α-Al2O3 (red lines) and α-Fe2O3 (green lines) agree very well, which indicates that the dislocation models applied in bulk materials before can be adapted to the misfit dislocations in this study. This agreement may be attributed to two reasons: firstly, the Poisson's ratios of sapphire and hematite are very close; secondly, the strain fields are not sensitive to the Poisson's ratio values according to Eqs. (8)–(9). By the comparison between experimental results (black lines) and simulation results, it can be concluded that the Foreman model with a = 2

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Fig. 4. (a) High-resolution TEM image of the α-Fe2O3/α-Al2O3 interface viewed along [1120]sapphire, inserted by Fourier transform of image with the analyzed spot circled for GPA. (b) The Fourier filtered image of (a). The inset is a magnified HRTEM image of one of the misfit dislocations.

best describes the strain fields of the misfit dislocations in the α-Fe2O3/ α-Al2O3 interfaces imaged along the [1100] direction. 3.2. Misfit dislocations imaged along the [1120] direction Fig. 4(a) shows a HRTEM image of another cross-section specimen, in which the cross-section plane is parallel to (1120) of sapphire as indicated by the inserted Fourier transform of Fig. 4(a). This image has been treated using the Fourier filter method to reduce the noise, and the result is shown in Fig. 4(b). An array of misfit dislocations is observed along the interface. The dislocation array is non-periodic and the spacing between misfit dislocations ranges from 7 nm to 9 nm. The inset in Fig. 4(b) shows a magnified HRTEM image of one of the misfit dislocations. The misfit dislocation core is characterized by one extra (1102) plane and one extra (1104) plane on the α-Al2O3 side of the interface and these extra half planes terminate at the interface. So its dislocation core structure is more complicated than the dislocation discussed in Section 3.1. By drawing a Burgers circuit surrounding the misfit dislocation core, the projected Burgers vector is determined to be a/2[1100]sapphire (~0.41 nm). This projected Burgers vector is parallel to the interface and accommodate the mismatch between the lattices of α-Fe2O3 and α-Al2O3 along the [1100] direction. The strain distribution around the dislocation core was investigated by GPA. As shown in Fig. 4(a), the (1102) and (1104) spots were chosen for the GPA and masks were placed around these spots to isolate them. The area between two adjacent dislocations was taken as the reference during GPA to obtain the optimal results for strain distribution [12]. Taking the x-axis parallel to [ 1 100] (along the interface) and the y-axis parallel to [0001], the strain distribution εxx, εxy, εyx and εyy can be calculated using GPA, as shown in Fig. 5(a)–(d). In the strain distribution along the x direction Fig. 5(a), it is interesting to find that there exist two strain fields around each misfit dislocation core. The two strain fields follow the same strain distribution characteristic to the dislocations discussed in Section 3.1: on the α-Al2O3 side, in which there exist the extra half planes, the strains are negative and compressive, and on the other side, the strains are positive and tensile; the strain is smaller farther from the dislocation core. The two strain fields are superposed and merged into each other to form the strain distribution around dislocation cores. The color scale indicates strain changes of −30% to +30%, higher than the strain of dislocation core in Section 3.1.

The dislocation density tensor was calculated from the strain distributions according to Eqs. (5)–(6). Fig. 5(e) and (f) shows the x-component αx and y-component αy of the dislocation density tensor, respectively. The bright/dark dots stand for the dislocation density distribution. Integrating the αx and αy over the whole dislocation core region we obtain the x-component bx (0.411 nm) and y-component by (almost zero considering the precision of 0.003 nm for GPA [18]) of the projected Burgers vector, which is in agreement with the value obtained by Burgers circuit. More importantly, integrating the two dots separately in αx component we obtain two Burgers vectors of 0.269 nm and 0.142 nm; in αy component, the two Burgers vectors have equal amplitude (0.216 nm) but opposite directions. In αx component, the dots with Burgers vectors of 0.269 nm are located at the left side of dislocation cores, while the dots with Burgers vectors of 0.142 nm are located at the right side. It can be seen that all the dislocations have the same Burgers vector in Fig. 5(e) and (f). However, the space distributions of dislocation density are quite different: in Dislocations 1, 3 and 4, as numbered in Fig. 5(a), the dots with Burgers vectors of 0.269 nm are located below the dots with Burgers vectors of 0.142 nm, while in Dislocation 2, the situation reverses. A detailed discussion on the space distributions of dislocation density will be presented next. By the dislocation density tensor analysis, we can conclude that the existence of two strain fields around the dislocation core, as discovered in Fig. 5(a), can be attributed to the two Burgers vectors corresponding to two dots in dislocation density distribution. Dislocation density tensor analysis is indeed superior to Burgers circuit analysis when the detailed dislocation core structures need to be investigated. A magnified αx and αy of one dislocation and the corresponding 3D visualization are shown in Fig. 5(g) and (h), respectively. It is observed that the dislocation density distribution is confined to the bright/dark dots (a pixel with 0.022 nm × 0.022 nm) and the whole region around these dots takes almost zero values (less than 1 × 10−6). The shape of αx and αy is a spike with its height corresponding to the projected Burgers vector, which indicates that the dislocation density tensor follows the classical elasticity theory, the same to the dislocation in Section 3.1. The dislocation density distributions are superposed on the strain distribution of two representative Dislocation 2 and Dislocation 3, as shown in Fig. 6(a)–(d). In both x-component and y-component, the locations of dislocation density distribution, masked by red/green dots, are in consistent with the maximal deformation areas of strain

Fig. 3. (a) The P–N dislocation model strain field, and (b–f) the Foreman dislocation model strain fields with a = 2, 3, 4, 5 and 6, respectively. (g) Experimental εxx strain distribution with overlaid circles with radii of 0.3 nm and 0.6 nm, from which line profiles are obtained. (h, i) The Foreman model (a = 2) strain field calculated using the Poisson's ratio of α-Al2O3 and αFe2O3, respectively. All the figures have the same color scale as inserted in (a). (j) Comparison of the circular line profiles taken from the experimental εxx strain distribution (black lines) and simulated Foreman model (a = 2) strain fields calculated using the Poisson's ratio of α-Al2O3 (red lines) and α-Fe2O3 (green lines) with different radii (solid lines for r = 0.3 nm and dash lines for r = 0.6 nm).

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Fig. 5. (a–d) The strain distribution εxx, εxy, εyx and εyy around the misfit dislocations obtained by GPA. The dislocations are numbered as shown in (a). All the figures have the same color scale as inserted in (a). (e, f) The x-component αx and y-component αy of dislocation density tensor, respectively. (g, h) A magnified αx and αy of one dislocation and the corresponding 3D visualization, respectively.

distribution. In order to reveal the origin of the different dislocation density distributions between Dislocation 2 and Dislocation 3, the detailed atomic configuration in dislocation cores was investigated. To distinguish the extra half planes clearly, two HRTEM images Fourier-filtered by choosing the (1102) and (1104) diffraction spots are shown in Fig. 6(e) and (f), respectively. The dislocation density distributions are superposed on the HRTEM images for comparison. It can be seen that the terminals of (1102) and (1104) extra half planes in the dislocation cores correspond to red and green dots with Burgers vectors of 0.269 nm and 0.142 nm, respectively, which indicates that the two dots in the dislocation density distribution are induced by (1102) and (1104) extra half planes. Thus we can conclude that atomic configuration in dislocation cores is in agreement with the space distributions of dislocation density, and the different space distributions

of dislocation density can be attributed to the differences of atomic configuration in dislocation cores. The strain distribution around dislocation cores was calculated and compared to experimental results. Fig. 7(a) and (c) shows the experimental εxx and εyy strain distribution of Dislocation 3. The simulated strain distributions were modeled by superposition of two strain fields calculated using the Foreman model with a = 2, the Poisson's ratio ν = 0.29 of sapphire, the origin situated at the dislocation core, and Burgers vectors of 0.269 nm and 0.142 nm for εxx, 0.216 nm and −0.215 nm for εyy, respectively. Fig. 7(b) and (d) shows the simulated εxx and εyy strain distribution, respectively. The color scale is the same for both experiment and simulation. To analyze the degree of agreement between the experiment and theory in more detail, the comparison of the experimental results and theoretical models was carried using circular profiles centered around the dislocation, as shown in

Fig. 6. (a–d) The strain distribution εxx and εyy of Dislocation 2 and Dislocation 3 with overlaid dislocation density distributions. (e, f) Two HRTEM images Fourier-filtered by choosing the (1102) and (1104) diffraction spots, respectively. The dislocation density distributions are superposed on the HRTEM images. The red/green filled dots coincide with the terminals of (1102) and (1104) extra half planes, respectively, and the red/green open dots are provided for reference.

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3.3. Two-dimensional network structure of misfit dislocations at the interface It should be realized that our experimentally observed projected Burgers vector component a/6[1120] along the [1100] direction and a/2[1100] along the [1120] direction is not a lattice translation. Dislocations and their characteristics, as well as their contribution to the plastic deformation behavior of the corundum type materials such as α-Al2O3 are well understood [35–37]. Energetically, a perfect basal dislocation with a Burgers vector a/3[1210] has the lowest energy; followed by the rhombohedral type dislocation with a Burgers vector of a/3[1010]. The prismatic type of dislocation with a Burgers vector of [1010] has the highest energy. It can be concluded that the observed misfit dislocations are both perfect basal dislocations with a Burgers vector a/3[1210]. In HRTEM image, only the projected Burgers vector (edge component) can be detected. As a result, when viewed along [1100], the projected Burgers vector is a/6[1120] = a/2; when viewed along [1120], the pffiffiffi projected Burgers vector is a/2[1100] = 3a=2. The dislocation line directions can't be determined from HRTEM images, since only the end view of a dislocation line is visible in the HRTEM image. Determination of the dislocation line direction can be carried out using conventional plan view TEM. However, as pointed out by Gutekunst and Wang [38,39], distinguishing the dislocation line orientation using conventional plan-view TEM contrast is precluded when the separation between dislocations is small, as in the present case (4.1 nm and 7 nm). However, our observation of different dislocation spacings along [1100] and [1120] clearly indicates that these dislocations do indeed form a two-dimensional network structure at the interface. 4. Conclusions In this work, a method, based on the combination of highresolution transmission electron microscopy, geometric phase analysis and dislocation density tensor analysis, was employed to investigate misfit dislocations in α-Fe2 O3 /α-Al2 O3 interface. The conclusions are as follows:

Fig. 7. (a, c) The experimental εxx and εyy strain distribution with overlaid circles with radii of 0.3 nm and 0.6 nm, from which line profiles are obtained. (b, d) The simulated εxx and εyy strain distributions. All the figures have the same color scale as inserted in (a). (e, f) Comparison of the circular line profiles taken from the experimental strain distribution (red lines) and simulated strain distribution (green lines) for εxx and εyy with different radii (solid lines for r = 0.3 nm and dash lines for r = 0.6 nm).

Fig. 7(a) and (c), for two radii r = 0.3 nm and 0.6 nm. The results are shown in Fig. 7(e) and (f) for εxx and εyy strain distribution, respectively. It can be seen that the simulated results are in good agreement with the experimental results, which proves that the Foreman model with a = 2 best describes the strain fields of the misfit dislocations, the same to the observation in Section 3.1.

1. The misfit dislocations form a two-dimensional network structure at the interface. When imaged along the [1100] direction, the misfit dislocation core is characterized by an extra (11 2 0) plane of α-Al2O3, referred to “Dislocation A”. When imaged along the [1120] direction, the misfit dislocation core is characterized by one extra (1102) plane and one extra (1104) plane of α-Al2O3, referred to “Dislocation B”. 2. The strain distribution around the dislocation core was investigated by geometric phase analysis. As for Dislocation A, there are no ycomponent strain fields (εyx and εyy) in the dislocation core. This can be attributed to the (11 2 0) extra half plane perpendicular to the interface, which can induce x-component displacement field ux only. As for Dislocation B, there exist two strain fields around misfit dislocation core. The two strain fields are superposed and merged into each other to form the strain distribution around dislocation cores. 3. The dislocation density tensor was calculated from the strain distributions, which gives very high accuracy in determining the Burgers vectors. As for Dislocation A, integrating the α x over the dislocation core region we obtain the projected Burgers vector (0.242 nm). As for Dislocation B, integrating the αx and αy over the whole dislocation core region we obtain the x-component bx (0.411 nm) and y-component by (almost zero) of the projected Burgers vector. More importantly, integrating the two dots separately in αx component we obtain two Burgers vectors of 0.269 nm and 0.142 nm; in αy component, the two Burgers vectors have equal amplitude (0.216 nm) but opposite directions. The two Burgers vectors induce two strain fields around dislocation core.

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4. The dislocation density tensor proved useful in accurately localizing the dislocation distribution in the core region. As for Dislocation B, it can be seen that all the dislocations have the same Burgers vector, however the space distributions of dislocation density are quite different. By dislocation density tensor analysis in combination with HRTEM observation, it can be seen that the two dots in the dislocation density distribution are induced by (1102) and (1104) extra half planes, from which we conclude that the different space distributions of dislocation density can be attributed to the differences of atomic configuration in dislocation cores. 5. The 3D visualizations of the dislocation density tensor corresponding to Dislocation A and Dislocation B both show a spike with its height corresponding to the Burgers vector, which indicates that the dislocation density tensors follow the classical elasticity theory. 6. The experimental strain distributions around the misfit dislocations were compared with the P–N and Foreman dislocation models. As for Dislocation A, the strain distribution is in agreement with a strain field calculated using Foreman model with a = 2. As for Dislocation B, the strain distribution is in agreement with the superposition of two strain fields calculated using Foreman model with a = 2 and the corresponding Burgers vectors. An identical dislocation strain model applied to both Dislocation A and Dislocation B demonstrates that they have the same spatial extension of strain field. As mentioned above, dislocation density tensor analysis is indeed superior to Burgers circuit analysis when the detailed dislocation core structures need to be investigated. The method in this work can be employed to investigate misfit dislocations in other system, particularly the systems with complicated core structure. Acknowledgments This research was supported by the Fundamental Research Funds for the Central Universities (N130402003). References [1] T. Fujii, D. Alders, F.C. Voogt, T. Hibma, B.T. Thole, G.A. Sawatzky, In situ RHEED and XPS studies of epitaxial thin α-Fe2O3 (0001) films on sapphire, Surf. Sci. 366 (1996) 579. [2] W. Weiss, M. Ritter, Metal oxide heteroepitaxy: Stranski–Krastanov growth for iron oxides on Pt(111), Phys. Rev. B 59 (1996) 5201. [3] Y.J. Kim, Y. Gao, S.A. Chambers, Selective growth and characterization of pure, epitaxial α-Fe2O3(0001) and Fe3O4(001) films by plasma-assisted molecular beam epitaxy, Surf. Sci. 371 (1997) 358. [4] S.I. Yi, Y. Liang, S. Thevuthasan, S.A. Chambers, Morphological and structural investigation of the early stages of epitaxial growth of α-Fe2O3 (0001) on α-Al2O3 (0001) by oxygen-plasma-assisted MBE, Surf. Sci. 443 (1999) 212. [5] S. Shara, J. Narayan, Strain relief mechanisms and the nature of dislocations in GaAs/ Si heterostructures, J. Appl. Phys. 66 (1989) 2376. [6] A.J. McGibbon, S.J. Pennycook, J.E. Angelo, Direct observation of dislocation core structures in CdTe/GaAs(001), Science 269 (1995) 519. [7] A. Vilà, A. Cornet, J.R. Morante, P. Ruterana, M. Loubradou, R. Bonnet, Structure of 60° dislocations at the GaAs/Si interface, J. Appl. Phys. 79 (1996) 676.

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