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Abstract
In high-resolution transmission electron microscopy (HRTEM) images of heterostructures, it is always difficult to accurately determine the interface position and identify dislocations in a large field of view at tens to hundreds of nanometers due to the small lattice differences. However, in the heterostructure, the determination of the interface position is the key to obtain the true mismatch stress/strain field of the interface. Due to the magnifying effect of the digital moiré method on small differences, digital moiré technology was applied to determine Ge/Si heterostructure interfaces and large-area identification interface dislocations in HRTEM lattice diagrams in this study. By optimizing the frequency and angle of the reference lattice, the interface and dislocation position are clearly and intuitively displayed. How to accurately determine the position of the heterostructure interface and the dislocation of the large-area recognition interface from HRTEM images are studied through simulation experiments. The results show that when the frequency of the reference lattice and the specimen lattice are close, and the angle between them is within 10°, the position of the heterostructure interface can be accurately and intuitively determined by the naked eye according to the distortion characteristics of the moiré fringe. When the frequency of the reference lattice is 0.7 to 0.9 times of the specimen lattice, and the rotation angle is within 8°, the visually clear crossover phenomenon of the moiré fringes is used for large-area identification of interface dislocations. Using the phase measurement interface position sensitivity can reach the Å level. Using the phase-shifting digital moiré method the strain field on the dislocation core at the Ge/Si heterostructure interface and the interface stress distribution were quantitatively analyzed. Compared with the Peierls-Nabarro dislocation model and the Foreman dislocation model, Foreman's variable factor α = 4 is more suitable for describing the strain field of misfit dislocations on the Ge/Si heterostructure interface.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Semiconductor heterostructure material is a new generation of semiconductor materials based on the molecular beam epitaxy and metal organic chemical vapor deposition [1,2]. Owing to its excellent photoelectric properties, it has been widely used in electronic, optoelectronic and energy conversion devices [3–5]. However, in the production of the heterostructure, due to the difference in the lattice constants, lattice mismatch is inevitable. The resulting strain energy accumulation would generate misfit dislocations or other defects at the film-substrate interface during the growth process [6–8]. The performance, yield strength, reliability, and degradation behavior of semiconductor devices would suffer from the presence of defects, especially dislocations, which can adversely affect its electronic performance [9]. At the same time, with semiconductor devices sizes decreasing into the nanometer scale, strain engineering in the Si channel becomes a critical part of improving performance for large scale integrated circuits. The presence of stress and strain in the interface film may also greatly improve the performance of the micro-nano electronic components. Accurate measurement of these complex interface strain and stress fields will provide a scientific basis for the optimal design of microelectronics and optoelectronic devices. The strain at more general heterophase boundaries containing interfacial dislocations has been treated within elasticity theory as a bimaterial of two joined half-spaces, each having the elastic properties of the bulk [10]. Therefore, it is especially important to determine the interface of the two-phase material during the calculation process, but the interface between two adjacent atomic phases is difficult to be accurately observed in high-resolution images. How to accurately measure the misfit dislocation strain field and interface strain/stress distribution at the interface of heterostructures has become a key issue. Moreover, there are many dislocation models for theoretical analysis of dislocations, such as the elastic theory model, the discrete model, the Peierls-Nabarro (P-N) dislocation model [11,12], the improved P-N dislocation model by Foreman [13], and so on. However, it is still not clear that which dislocation model is best suited for describing the misfit dislocation at the Ge/Si heterostructure interface.
The sampling moiré [14] and phase-shift digital moiré method [15] are widely used in deformation testing due to their fast and high precision. For example, S Ri [16] et al. used the sampling moiré method to accurately measure the small deformation distribution of the three-point bend of the steel beam. Some researchers have applied the sampling moiré method to the defect detection of the GaN atomic structure [17]. Masako Kodera [18] et al. developed a two-dimensional fast-Fourier-transform sampling moiré technique to visually and quantitatively determine the locations of minute defects in a transmission electron microscopy image. Some researchers used a phase-shifted moiré method to measure thermal deformation in a four-plane package electronic package under an atomic force microscope [19]. Some researchers applied the digital phase shift method to microscopic scale 3D deformation measurement [20] Optical moiré technique is widely used in related fields since 1948, which is still a powerful and promising technique for researchers. Some researchers have developed many new methods based on the principle of geometric moiré technology [21,22]. Some researchers use HREM crystal material images and unidirectional grating superposition to form a nano-moiré method with nanometer sensitivity and spatial resolution [23]. It can be used to measure important displacements, strains, dislocations, superposition faults and atomic bond ruptures in nano-mechanics. Some researchers used a TEM lattice image to generate a reference lattice and a sample lattice [24]. A digital rotating moiré strain measurement method based on HRTEM lattice image is proposed. The sampling moiré method, the digital phase shift method, and the digital moiré method described above are mostly used for deformation measurements that do not include an interface. Although, GPA is widely used in various deformation tests due to its high precision and strong noise immunity, such as strain field analysis close to dislocations [25–27]. However, the GPA method can only measure the strain of a single type of atomic lattice structure. For multilayer structures, the measurement results are almost all relative strain fields [28–32]. And during the GPA calculation process, the selection process involves the selection of the size and shape of the diffraction point and filtering methods, which may make the calculating result different by different operator [33]. However, in materials of semiconductor heterostructures, the real strain/stress at the interface is more important for mechanical characterization, especially for structural properties on both sides of the interface. How to accurately determine the interface of the heterostructure and correctly select the reference area of each part is the key problem of the actual interface strain/stress analysis.
In this study, the application of digital moiré has an amplification feature for small deformations. By precisely adjusting the frequency and angle between the reference grid and the sample lattice, the position of the interface in the high-resolution image of the heterostructure is clearly and intuitively displayed, and the position of the defect in the dislocation can be easily recognized on a large field of view. The simulation test results show that when the frequency of the reference lattice and the specimen lattice are close, and the angle between them is within 10°, the position of the heterostructure interface can be accurately and intuitively determined according to the distortion characteristics of the moiré fringes. When the frequency of the reference lattice is 0.7 to 0.9 times of the specimen lattice, and the rotation angle is within 8°, the visually clear crossover phenomenon of the moiré fringes is used for large-area identification of interface dislocations. And combined with the four-step phase-shifting method, the phase-shifting digital moiré method is used to accurately measure the strain field around a mismatching dislocation at the interface of the Ge/Si heterostructure and the interface stress distribution were quantitatively analyzed using elastic theory. The theoretical strain fields calculated by the P-N and Foreman dislocation models were analyzed and compared. The accurate measurement of the internal strain/stress distribution of the semiconductor heterostructure and the evaluation of the overall heterogeneous lattice quality provide an important basis for judging the optoelectronic performance of the whole device and the optimization design of the heterostructure.
2. Method principle
2.1 Digital moiré formation method
Moiré method is a relatively mature method of photomechanics, which uses two groups of the grating with periodic changes to generate moiré fringes. The moiré fringes will move macroscopically with a slight change in the geometry of the grating, where the movement of the moiré fringes is directly used for the displacement and strain fields analysis of the tested object. In the moiré method, the deformed grating is the carrier of the deformation information of the object. Therefore, the appropriate grating can be utilized to quickly and easily extract the deformation information of the object. In this study, the reference lattice is edited by MATLAB software to ensure there is not any unknown deformation in the reference lattice. That is, a high-resolution image of the crystal material (specimen lattice) is superimposed with an unidirectional digital lattice (reference lattice) to form a moiré fringe, which shows the difference between the specimen lattice and the reference lattice, and this difference contains information such as displacement, strain, and atomic defects. Therefore, the digital moiré method effectively reduces the possible experimental error. Simultaneously, the reference lattice can be accurately moved in the computer, and the reference lattice frequency and the angle between the reference lattice and the specimen lattice can be easily adjusted. Moiré fringes with different frequencies and angles can be obtained. Interface position can be quickly determined in heterostructure. And defect locations such as dislocations can be conveniently identified over a large field of view.
In this study, an HRTEM image of Ge/Si heterostructure (11-1) crystal face were used as the specimen lattice. The intensity distribution of the periodic specimen lattice can be expressed as
where Ns0 is the mean background intensity, Ns1 is the modulation amplitude, and ps is the pitch of the specimen lattice, φs is the phase component that carrying the deformations.The reference lattice was generated by MATLAB programming. The intensity distribution of the periodic specimen lattice and reference lattice can be expressed as
where Nr0 and Nr1 are the constants related to the mean background intensity and modulation amplitude, respectively, and pr is the pitch of the reference or specimen lattice. φr is the initial phase.According to the moiré formation principle, when the reference lattice is superimposed with the specimen lattice, a clear moiré fringe is produced. The intensity distribution function can be calculated by
The stress is obtained by the following equation:
where E is the Young's modulus; v is the Poisson's ratio, ${\varepsilon _x}$ and ${\varepsilon _y}$ are strains in the x and y directions, respectively.2.2 Method for judging the position of the interface position
The two semiconductors that normally form a heterojunction have the same crystal structure and close atomic spacing. At the same time, any one of the crystal directions in the HRTEM image can be regarded as a periodic orthogonal lattice. The direction and spacing of the moiré fringe are related to the frequency of the reference lattice and the included angle (θ) between reference lattice and specimen lattice. Due to the different lattice spacing at the ends of the heterostructure interface, there are two different fringes at the ends of the interface. When the reference lattice is rotated in the plane, the moiré spacing (δ), the angle (β) between the main direction of the moiré and the horizontal direction can be expressed as
The angle ($\Delta \beta $) between the moiré fringes at both ends of the interface can be obtained from Eqs. (11) and (12).
The rotation of the moiré fringes was generated by simulation experiments, which verified the derivation of the above theoretical equation. Figure 1(a) is a simulated orthogonal lattice (as a specimen lattice) image with different frequency (the frequencies are 0.136 and 0.130 respectively) using MATLAB software. In order to observe the variation of the moiré fringe more intuitively, series moiré patterns are produced, as shown in Figs. 1(b)–1(d). The red dashed line is the contact interface of two orthogonal lattice. The frequency of the reference lattice changes from 0.115 to 0.135, with an interval of 0.01. The angle between the reference lattice and the sample lattice is constant at 5° (θ1=θ2 =θ). As the frequency of the reference lattice approaches the specimen lattice (f1 = 0.136, f2 = 0.130), the moiré fringe is bent at the interface (see the red dotted line). Therefore, the simulation test results are consistent with the inference of Eq. (15). As shown in Figs. 1(f)–2(h), the angle between the reference lattice and the specimen lattice varies from 1° to 15°. The frequency of the reference lattice is constant at 0.13. As the angle decreases, the moiré fringe width becomes wider and wider, and the bending angle at the interface becomes larger and larger. The simulation results verify the results of Eq. (16). The simulation test illustrates that when the frequency of the reference lattice and the specimen lattice are close, the bending of the moiré fringe at the interface can be observed with the naked eye at an angle of 10°, which is used to judge the position of the interface.
Fig. 1. Rotating moiré, (a) simulated lattice structure with different frequency; (b) θ = 5°, fr = 0.115; (c) θ = 5°, fr = 0.125; (d) θ = 5°, fr = 0.135. (e) θ = 1°, fr = 0.130; (f) θ = 5°, fr = 0.130; (g) θ = 10°, fr = 0.130 ; (h) θ = 15°, fr = 0.130.
Figure 2(a) is a four-step phase shift fringe obtained by Gaussian filtering, and the phase shift amounts are 0,π/2, π, and 3π/2, respectively. Figure 2(b) shows the phase field after wrapped using a four-step phase shifting. Figure 2(c) shows the gradient line obtained by the phase field along the stripe direction. If the fringes are not changed, the gradient lines obtained by the phase field are continuous line segments. When the moiré fringe changes in bending, the continuity of the gradient line at the bend point changes. Therefore, the interface position can be accurately determined according to the continuity of its phase field gradient line. This method improves the accuracy and sensitivity of measurement and realizes automatic measurement. The accuracy of the phase is one percent of the displacement, so the theoretical accuracy of the phase shift method used to determine the position of the interface can reach the Å level.
Fig. 2. (a) Four-step phase-shifted moiré fringe patterns extracted from Fig. 1(d); (b) wrapped phase field; (c) gradient line obtained from the phase field.
2.3 Method for identifying dislocations over a large area
High-resolution transmission electron microscopy (HRTEM) is commonly used to observe dislocation information at the heterostructure interfaces. However, it is difficult to quickly identify dislocations in large areas in HRTEM images, since the lattice anomalies caused by dislocations are only within a few atomic ranges in thousands of atoms. In this paper study, the digital moiré method is applied to quickly identify dislocations in a large area in the HRTEM image of heterostructure. We used software simulation to generate three common dislocations in HRTEM images, as indicated by the arrows in the white boxes in Fig. 3. It is found through experiments that when there are dislocation defects in the crystal, the misalignment phenomenon of the moiré fringes caused by the sliding of atoms in different rows (as shown in the white box of Fig. 3(b)). At the same time, the intersection of the moiré fringes is always at the dislocation position and does not change with the angle, as shown in the white box of Fig. 3(c). So, there is an intersection after the superposition and the intersection location does not change with the rotation of the reference lattice, we can determine the existence of the dislocation defect. It is found through experiments that when the frequency of the reference lattice is 0.8 ∼ 0.9 times of the specimen lattice, and the angle is within 10°, the formed moiré fringes are suitable for large-area recognition of dislocations in HRTEM images. That is, a high-resolution image of the crystal material (specimen lattice) is superimposed with an unidirectional digital lattice (reference lattice) to form a moiré fringe, which shows the difference between the specimen lattice and the reference lattice, and this difference contains information such as displacement, strain, and atomic defects.
Fig. 3. (a) Three white boxes A, B, and C are the dislocation locations generated by the software, (b) and (c) different angles of rotational moiré fringes formed by the reference lattice and the specimen lattice, the white square is the location of the moiré fringe intersection.
3. Experiment and discussion
An ultra-high vacuum chemical vapor deposition system equipped with pyrolytic BN effusion cells was used to grow Ge films on Si (001) substrate. The TEM specimens were prepared for cross-sectional imaging along the [1–10] direction using a standard technique, which involved mechanical grinding followed by ion milling. An HRTEM experiment was performed on a JEM-2010 transmission electron microscope at 200 kV.
3.1 Application of digital moiré method in the determining Ge/Si heterostructure interface position
Figure 4(a) is an HRTEM image of a Si/Ge heterostructure in which the yellow region is the interface between Si and Ge, and the two sides are single crystal Ge and single crystal Si regions, respectively. In the HRTEM image, the position of the interface can only be roughly judged. To accurately determine the position of the interface, it is necessary to continue to develop new methods. In the paper, since the lattice constants of Si (a = 0.5428 nm) and Ge (a = 0.5658 nm) are different in the Ge/Si heterostructure, the (11-1) interplanar crystal spacing is also different. Through simulation experiments, we found that the moiré fringes formed by superimposing with the reference lattice when there are two frequencies in the specimen lattice have bending phenomenon, and the line connecting the bending points is the interface between the two frequencies in the specimen lattice. In this paper, moiré fringe images of various angles were generated, in which the best result was obtained when parallel moiré is formed between the reference lattice (fr = 0.130) and the specimen lattice (the (11-1) crystal face of the Ge/Si heterostructure). There is an obvious bending point in the moiré fringes. Therefore, the bending points of moiré fringe are sequentially connected (see the red dashed line in Fig. 4(c)), and the interface position of Ge/Si heterostructure can be determined. The red dotted line in Fig. 4(d) is the interface position determined in the HRTEM image using the above method. It is stated here that the interface determined by this method is not an interface in the strict physical sense, but is the middle position of the transition area between the two substances.
Fig. 4. (a) HRTEM image of Si/Ge heterostructure, the yellow area is the interface between Si and Ge; (b) reference lattice parallel to the Si/Ge heterostructure (11-1) crystal face; (c) interface on moiré fringe; (d) real interface on Si/Ge heterojunction HRTEM image.
3.2 Digital moiré method to quickly identify misfit dislocations on the heterostructure interface
In diamond crystal structures, such as Si, Ge and Si1-xGex, dislocations mainly propagate in the {111} crystal face, so misfit dislocations are mainly located in the {111} crystal face near the interface [34]. Figure 5(a) shows the HRTEM image of Ge/Si heterostructure. Figure 5(b) shows a rotation moiré image formed by superimposing the (11-1) crystal face of the HRTEM image with the reference lattice (p = 9 pixels) after a 9° (the angle between the reference lattice and the specimen lattice) counterclockwise rotation. The intersection of fringes can be clearly seen in the four white rectangular boxes region of A, B, C, and D. This phenomenon is very similar to the presence of misfit dislocations in the Ge/Si heterostructure, so it is determined that there may be misfit dislocations. Four white squares A, B, C, D with the same size and position are also marked in the HRTEM image (see Fig. 5(a)). Four misfit dislocations on the (11-1) crystal face of the Ge/Si heterostructure was observed in the squares, as indicated by the white arrow in the Fig. 5(a). There is only one dislocation in each square of this direction, which is corresponding to the change of moiré fringes in Fig. 5(b). Therefore, it can be determined that the intersection of moiré fringes in Fig. 5(b) is caused by misfit dislocations on the interface, and the position information of misfit dislocations in the Ge/Si heterostructure can be quickly and accurately captured by the digital moiré method.
Fig. 5. (a) HRTEM images of Ge/Si heterostructure; (b) rotation moiré formed by the reference lattice and the specimen lattice.
3.3 Application of sampling moiré method in determination of burgers vector
Figure 6(a) is the interface region intercepted in the HRTEM image of Si/Ge heterostructure. To facilitate the calculation, the interface is rotated to the horizontal direction where the red dashed line is the interface position of the Ge / Si heterostructure. The upper region is Ge, and the lower is Si. Figure 6(b) shows the Fourier transform of Fig. 6(a), in which the analyzed spots have been encircled. Usually, the type of these misfit dislocations can be directly determined from the HRTEM images by drawing a Burgers circuit around the dislocation. It is observed from Fig. 6(a) that the lattice atoms in the HRTEM image are not obvious, which make it difficult to determine the type of dislocation for drawing the Burgers circuit. For this reason, the sampling moiré method, with the advantages of large field of view and a significant amplification of the distortion, was used to identify the Burger circuit, where the pixels were acquired by sampling in the horizontal and vertical directions with a constant pixel interval in the HRTEM image respectively. The sampling interval approximates the interplanar crystal spacing of Ge/Si heterostructure in the horizontal (13 pixels) and vertical (11 pixels) directions, respectively. Figure 6(c) shows the sampling moiré image of Fig. 6(a), in which the misfit dislocation core in the Ge/Si interface can be seen clearly. The Burgers vector is b = 1/2 [110].
Fig. 6. (a) HRTEM image of Ge/Si heterostructure interface; (b) Fourier transform of (a); (d) burgers vector with sampling moiré.
3.4 Strain field of dislocation core at Ge/Si interface
Then, the digital moiré method is used to characterize the strain field around a misfit dislocation. Figure 7(a) calculated region is shown in the white rectangle in Fig. 6(a). It is very convenient to realize the phase shift by moving the reference lattice phase. Taking the parallel moiré pattern as an example, the reference lattice is superposed with the specimen lattice (the (11-1) crystal face of the Ge/Si heterostructure). When the four reference lattices are superposed with the same specimen lattice, the four-step phase-shifted moiré pattern can be obtained, and the phase shift amounts are 0, π/2, π, and 3π/2, respectively, as shown in Fig. 7(b). Figure 7(c) is a moiré pattern extracted by filtering from Fig. 7(b).
Fig. 7. (a) HRTEM image of Ge/Si heterostructure interface; (b) digital moiré fringe patterns; (c) fringe patterns extracted from (b).
Figure 8 shows the strain maps around the misfit dislocations at the Ge/Si heterostructure interface. The strain maps of the misfit dislocations are all eight-shaped, with the strain value ranging from -8% to + 8%. The experimental ɛxx strain field calculated using the four-step phase shift method is shown in Fig. 8(a). Figure 8(b) shows the strain component ɛxx calculated using the P-N dislocation model [25]. Figures 8(c)–8(k) demonstrate the strain components ɛxx given by the Foreman model [10], with the Poisson's ratio νSi = 0.278, Burgers vector bSi = 0.384 nm, Young's modulus is 130 GPa; Poisson's ratio νGe = 0.32, bGe = 0.4 nm, Young's modulus is 79.9 GPa, and the corresponding alterable factors10 ranging from α = 2 to α = 10. The eight-shaped strain maps become wider and the dislocation width increase with the increase of the factor α in the Foreman model. There exists a convergence region around the dislocation core. The maximum strain value appears in the core area of the dislocation and the strain value become smaller away from the dislocation core. The strain map of the Foreman model (α = 4) is qualitatively similar to the experimentally determined strain in Fig. 8. Thus, the Foreman model with α = 4 best describes the strain fields of the misfit dislocation core in the Ge/Si heterostructure interface.
Fig. 8. Experimental and theoretical strain fields: (a) experimental strain field; (b) strain field of P-N dislocation model; (c) - (k) strain field of Foreman dislocation model, with α ranging from 2 to 10, respectively.
3.5 The strain and stress distribution near the interface of dislocation
At the center of dislocations, atoms are arranged in a disorderly manner and beyond the range of elastic deformation. Hooke's law cannot be applied in the dislocation core, but away from the center area the Hooke’s law can be still used for elastic deformation analysis. In this paper, Ge/Si heterostructure are separated at the interface and treated as two materials with two connected half-spaces, and each half-space is considered to be within the elastic deformation range so that the interface stress can be calculated through elastic deformation.
Figure 9(a) is an interface area extracted away from the dislocation core. The red dashed line is the center of the Ge/Si heterostructure interface. The corresponding parallel moiré fringes are obtained by superimposing specimen lattice (the (11-1) crystal face of the Ge/Si heterostructure) and the reference lattice. Taking the phase far away from the interface as the reference zone, the corresponding displacement ur and un can be obtained by the phase shifting method. Then, the displacement in the horizontal and vertical directions can be obtained using ux= urx + unx and uy= ury + uny, respectively. The strain fields of the Ge/Si heterostructure in the horizontal is shown in Fig. 9(b). There exists a convergence region of the strain field around the interface. The maximum strain value appears in the interface area and the strain value become smaller away from the dislocation core. The upper Ge region is compressively strained and lower Si region is tensile strained. This is due to the fact that Ge (a = 0.5658 nm) has a larger lattice constant than Si (a = 0.5428 nm). When Ge and Si are combined at the interface, due to the difference in lattice constant, the upper Ge is compressed by the Si with a smaller lattice constant. Conversely, the lower Si is stretched. Then, stress is calculated suing the elastic theory.
Fig. 9. Interfacial strain and stress of Ge/Si heterostructure: (a) intercepted Ge/Si heterostructure interface; (b) strain field in the horizontal direction; (c) stress profile corresponding to the dash-dotted line in (b).
Figure 9(c) shows the stress profile of the Ge/Si heterostructure corresponding to the dash-dotted line in Fig. 9(b). The stress approaches 0 at a distance away from the interface in the Ge region, and the compressive stress become larger when getting closer to the interface, with a maximum value of 3 GPa. When passing through the interface to reach the Si region, the compressive stress becomes tensile stress and reaches a maximum of 6.6 GPa near the interface.
4. Conclusions
In this study, digital moiré techniques were used to accurately determine the interface of heterostructures and large-area recognition dislocation defects in HRTEM lattice images. The experimental results show that the digital moiré method magnifies the small deformation in the HRTEM image, and can visually observe the existence of interface misfit dislocations and mismatched strains in the moiré pattern. When the frequency of the reference lattice and the specimen lattice are close, and the angle between them is within 10°, the position of the heterostructure interface can be accurately and intuitively determined according to the distortion characteristics of the moiré fringes. When the frequency of the reference lattice is 0.7 to 0.9 times of the specimen lattice, and the rotation angle is within 8°, the visually clear crossover phenomenon of the moiré fringes is used for large-area identification of interface dislocations. The digital phase shifting method is used to characterize the strain field distribution near the dislocation. Compared with the P-N dislocation model and the Foreman dislocation model, Foreman's variable factor α = 4 is more suitable for describing the strain field of misfit dislocations on the Ge/Si heterostructure interface. Simultaneously, quantitative analysis is made on the strain and stress distribution of the interface away from the dislocation, which provides a scientific basis for the optimal design of microelectronics and optoelectronic devices.
Funding
National Natural Science Foundation of China (11232008, 11372037, 11572041).
Acknowledgments
The authors are grateful to all of those who were involved in this work
Disclosures
The authors declare no conflicts of interest.
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