TY - JOUR
AU - Watanabe,, Kazuto
AB - Abstract The quantitative measurement of a crystal bending effect is performed using low-order zone-axis convergent beam electron diffraction (CBED) patterns. Although the accuracy of the present method is inferior to that of the method of using split higher order Laue zone lines, this method enables us to estimate the crystal bending effect at a region very close to the interface and to easily judge whether the crystal bending effect results in a tensile bend or a compressive bend. As an application of the present method, the crystal bending effect at a region close to the SiGe/Si interface was measured. It was found that the crystal bending effect is due to a thin-foil relaxation of almost 0.3° at a region that is ∼10 nm away from the interface. crystal bending effect, CBED, dynamical simulation, split HOLZ lines, SiGe Introduction Recently, there has been a strong demand for a device evaluation method that has a high spatial resolution because the 90-nm technology node is now in practical use. In particular, the presence of a strain in the transistor channel region is very important to progress or deteriorate a device property. Transmission electron microscopy (TEM) is one of the powerful techniques for analysing a region of interest at a high spatial resolution. In practice, a convergent beam electron diffraction (CBED) method is widely used for measuring the local lattice parameters with high accuracy [1,2]. The CBED method obtains an electron diffraction pattern by illuminating a fine convergent electron beam onto a specimen. Diffracted discs appear at the positions of a Bragg reflection, and dark lines that are known as higher order Laue zone (HOLZ) lines appear in the central disc. The lattice constants can be measured precisely by comparing the experimental and kinematically simulated HOLZ lines [3–11]. However, the distortion induced by a high-stress field complicates the CBED pattern due to the appearance of split HOLZ lines. This makes the CBED measurement difficult. Banhart [12] explained that the split HOLZ lines are caused by a strain relaxation in the thin-foil TEM lamella, due to a difference in the thermal expansion coefficient between the matrix and the cover layer. Recently, Chuvilin et al. [13] performed a simulation for the quantitative analysis of the crystal bending effect using a multi-slice algorithm. Furthermore, Soeda [14] performed the quantitative analysis by comparing the experimental split HOLZ lines with the dynamically simulated results based on the three-dimensional Bloch wave theorem [15,16]. Very recently, the crystal bending effect of an actual device was estimated by a novel approach based on a kinematical simulation [17]. In the analysis using the split HOLZ lines, the crystal bending effect was measured from the splitting line width, and the accuracy of the measurement was very high because of the clear criterion. However, it was slightly difficult to measure the crystal bending effect at a region very close to the interface because the split HOLZ lines became indistinct due to a sharp crystal bend. In addition, it was impossible to judge whether the crystal bending effect resulted in a tensile bend or a compressive bend. Furthermore, in order to obtain a clear HOLZ line, the crystal had to be inclined to ∼10° from a low-order zone axis. The correct crystal bending effect against a cover layer has not yet been measured. In this paper, in order to compensate for these faults, the quantitative measurement of a crystal bending effect by using a low-order zone-axis CBED pattern based on the zeroth-order Laue zone (ZOLZ) reflections is proposed. The quantitative measurement is performed by carrying out a two-dimensional pattern fitting between the experimental low-order zone-axis CBED patterns and the dynamically simulated patterns including the crystal bending effect. In general, the strain effects induced by a stress are classified into two parts: a lattice strain and crystal bend. Although it is very important to simultaneously measure the lattice strain and bend effect, it very difficult to measure the lattice strain using only ZOLZ reflections. Therefore, we first focused to measure the crystal bending effect. Method Dynamical simulation including the crystal bending effect The schematic diagram of a compressive crystal bend is shown in Fig. 1a. However, it is very difficult to simulate this external form of the crystal exactly. Therefore, we ignore the displacement of the vertical component to the surface of the incident beam due to the bend and develop a structural model that considers only the displacement of the tangential component to the surface, as shown in Fig. 1b [18,19]. Because a CBED pattern formed by the ZOLZ reflections is mainly an object of this study, the error due to ignoring the vertical components hardly influences the final result. Fig. 1 Open in new tabDownload slide (a) Schematic diagram of an exact crystal bend. (b) Schematic diagram of the crystal bend excluding the displacement of the vertical component to the surface. (c) Schematic diagram of the crystal bend utilized in the layer-by-layer simulation. Fig. 1 Open in new tabDownload slide (a) Schematic diagram of an exact crystal bend. (b) Schematic diagram of the crystal bend excluding the displacement of the vertical component to the surface. (c) Schematic diagram of the crystal bend utilized in the layer-by-layer simulation. In order to calculate the crystal bending effect as shown in Fig. 1b, a fully dynamical n-beam theorem based on the three-dimensional Bloch wave method is expanded by taking into account the propagation of a fast electron wavefunction through multiple layers [15,16]. A brief outline of this algorithm is given below. In general, the transmission coefficient of the multiple layers constructed by the n-layer is given by (1) where Cm(K) is an eigenvector of the mth layer; K is the tangential component to the surface of the incident wave number; tm is the thickness of the mth layer; Γ(tm) is a diagonal matrix of [Γm(tm)]j,j = exp(iγm(j)tm) and γm(j) is the ‘anpassung’ of the ith state in the mth layer. The passage of the fast electron wavefunction through the bent layers is expressed by an origin shift of the unit cell between multiple layers. In other words, the crystal bending effect is equivalent to the change in the direction of the fast electron propagation. A schematic diagram of the crystal model utilized in the layer-by-layer simulation is shown in Fig. 1c. In Fig. 1c, the crystal bend is expressed by the origin shift of a unit cell described by the bold square. This simplified model is a good approximation to develop the model shown in Fig. 1b if each layer is set to a sufficiently thin layer. The origin shift of the unit cell can be easily introduced into Eq. (1) as follows: (2) where D(δm) is a diagonal matrix of [D(δm)]j,j = exp(ig·δm); g is the reciprocal lattice vector and δm is the position vector expressing the origin shift of the unit cell at the mth layer. The split HOLZ lines of Si(210) simulated by different quantities of the crystal bending effect are displayed in Fig. 2a–f. The crystal bending effect is induced along the [001] direction. Here, the bending lattice plane is introduced as a quadratic function of the specimen thickness such that the crystal bending effect becomes maximum at the central position of the specimen thickness. In a crystal bending effect assumed in this way, Uesugi et al. [17] defined the quantity and the direction of the maximum displacement as the bending vector Δ. By following this notation, the one-dimensional external form of a bended lattice plane is expressed by y = Δ(t/2)−2x2, where t is the specimen thickness, and x and y coordinates are defined in reference [17]. Δ is used as scalar in this paper because the shape of the crystal bend is very simple in the present sample as later mentioned. This quantity makes it convenient to introduce the crystal bending effect into a dynamical simulation. On the other hand, the crystal bending effect can also be defined as the bending angle θ, as shown in Fig. 1b [12]. This definition results in the crystal bending effect becoming independent of the sample thickness. In this study, we carefully choose between both these notations according to the circumstances. In these notations, the positive quantities of the crystal bending effect indicate a tensile bend and the negative ones indicate a compressive bend. Fig. 2 Open in new tabDownload slide (a–f) Simulated CBED patterns of [210]-orientated Si at each bending vector. (g–l) Simulated CBED patterns of []-orientated Si at each bending vector. Fig. 2 Open in new tabDownload slide (a–f) Simulated CBED patterns of [210]-orientated Si at each bending vector. (g–l) Simulated CBED patterns of []-orientated Si at each bending vector. Figure 2 shows that the width of the split HOLZ lines increases with an increase in the magnitude of the bending vector, and the whole contrast of the central disk becomes obscure as the magnitude of the bending vector increases. On the other hand, the low-order zone axis CBED patterns of Si(⁠⁠) simulated by the respective bending vectors are displayed in Fig. 2g–l. It is found that all the CBED patterns are asymmetric against the direction considering the crystal bending effect. Nevertheless, it is important that the symmetry of the central disk hardly change. This fact proves that the asymmetrical character of all the patterns is not caused by a crystal tilt. Furthermore, the entire contrast of the CBED pattern is distinct when compared to that of the split HOLZ lines even when the same the bending vectors as those shown in Fig. 2a–f are used for the dynamical simulation. These results lead us to the conclusion that we can measure the bending vector in a high-stress region where it is difficult to measure the bending effect by using the split HOLZ lines. Experimental procedures Commercial SiGe wafers with a (001) surface were used in this experiment. They comprised a 50-nm-thick Si0.8Ge0.2 layer epitaxially grown on a Si(001) substrate. A cross-sectional sample was prepared as follows. The SiGe wafers were bonded by epoxy glue onto each other. Then, the samples were mechanically thinned and polished to a thickness of ∼20 μm by using a dimple grinder. The final thinning was carefully carried out by means of 4-keV Ar+ ion milling at an angle of 4°. CBED patterns were obtained by using a HITACHI H-9000NAR electron microscope operated at 100 kV with a 10-nm probe and an LaB6 filament. All measurements were performed at room temperature. A high-resolution TEM (HRTEM) image and a high-angle annular dark-field scanning TEM (HAADF STEM) image, which were used for evaluating the SiGe wafers, were obtained by using a JEOL JEM-2100F TEM/STEM electron microscope with an atomic-resolved fine probe and a field emission gun. Sample property Before the sample thickness and crystal bend are examined, the influences of a lattice strain and an impurity concentration on the CBED patterns are discussed. In the measured region of the present experiment, whether these parameters should be taken into account was judged by using the HRTEM and HAADF STEM methods. Firstly, we examined the influence of the residual Ge atoms in a matrix region by using the HAADF STEM method. This method has a unique property called Z contrast, which means that the image intensity depends on the atomic number of elements forming the atomic column [20–22]. Figure 3a shows a HAADF STEM image of the []-orientated SiGe at 200 kV. The HAADF STEM image includes the boundary between a covered layer of SiGe and the matrix Si. The position of the boundary is pointed by the black arrows in Fig. 3a. From the image contrast, it is concluded that the region on the left-hand side is SiGe and that on the right-hand side is Si. Since almost uniform intensity is obtained in the Si matrix, the residual Ge atoms in the matrix region hardly influence the CBED patterns. Fig. 3 Open in new tabDownload slide (a) High-resolution HAADF STEM image of the SiGe/Si interface. (b) HRTEM image taken at the almost same region. (c) {200} phase map derived from the HRTEM image shown in (b). Fig. 3 Open in new tabDownload slide (a) High-resolution HAADF STEM image of the SiGe/Si interface. (b) HRTEM image taken at the almost same region. (c) {200} phase map derived from the HRTEM image shown in (b). In general, when the epitaxial growth is formed between the covered layer with a large mismatch of the lattice parameter and the substrate, a lattice strain is induced in the covered layer along the growth direction. Therefore, it is necessary to investigate the influence of the lattice strain. In this study, the geometric phase method suggested by Hÿtch et al. [23,24] is used for viewing a lattice strain easily. Figure 3b displays an HRTEM image of the []-orientated SiGe at 300 kV. It is confirmed that the 002 spot appearing in the Fourier diffractogram of this HRTEM image is slightly spread due to an induced strain. In order to obtain the phase map, the 002 spots of the Si and SiGe layer are selected. In general, the strain map is derived from two independent vectors. However, in the present sample, we assumed that there was no strain component along the [110] direction. Therefore, we directly relate phase map and strain. These selected spots are moved to another position as the 002 spot of Si is located at centre in Fourier space. Then, a phase map, shown in Fig. 3c, is obtained by performing an inverse Fourier transform. In order to clarify the change of phase, a phase wrapping is performed between 0 and π/2. The gradient of the phase map expresses the lattice strain in this method. In addition, the direction of the gradient shows the components of the strain. Consequently, it is confirmed that the lattice strain along the [001] direction is introduced into the SiGe layer. On the other hand, the phase wrapping is hardly observed in the Si layer. By using dynamical simulation, it is confirmed that CBED patterns based on the ZOLZ reflections are hardly changed by the small lattice strain in the Si layer of the present sample. Therefore, it is concluded that the strain in the Si layer is not large enough to have an influence on the ZOLZ reflections of the CBED patterns. Results and discussion A low-magnified bright-field (BF) image including a SiGe interface is shown in Fig. 4a. The black contrast corresponds to the SiGe layer. It is confirmed that the depth of the SiGe layer is ∼50 nm. The typical CBED patterns from the positions marked ‘A–D’ in Fig. 4a are shown in Fig. 4b–e, respectively. The CBED patterns become asymmetrical as the observed points approach the interface. Fig. 4 Open in new tabDownload slide (a) Low-magnification BF image including the typical observed points. (b–e) Experimental [] zone-axis CBED patterns at the regions marked by ‘A–D’, respectively. (f–i) Maps of the χ2 values with respect to the sample thickness and bending vector corresponding to (b)–(e), respectively. (j–m) Simulated CBED patterns at the minimum points in each map. Fig. 4 Open in new tabDownload slide (a) Low-magnification BF image including the typical observed points. (b–e) Experimental [] zone-axis CBED patterns at the regions marked by ‘A–D’, respectively. (f–i) Maps of the χ2 values with respect to the sample thickness and bending vector corresponding to (b)–(e), respectively. (j–m) Simulated CBED patterns at the minimum points in each map. In order to determine the bending vector of the crystal bending effect, a two-dimensional pattern fitting procedure was carried out between the experimental CBED patterns and the ones simulated with respect to two parameters, the bending vector and the sample thickness. Since the Kossel–Möllenstedt fringe is formed by the interference effect among the ZOLZ reflections of the changes in the CBED patterns caused by the sample thickness sensitivity, the sample thickness is also considered as one of the parameters that are necessary for the fitting procedure. In practice, although the four typical experimental CBED patterns were taken in a region of uniform thickness, it should be noted that the patterns of the central disk change. From this fact, it is also confirmed that sample thickness is an indispensable parameter for an exact pattern fitting. The dynamical simulation was performed for sample thicknesses ranging from 50 to 250 nm at intervals of 5 nm and for the bending vectors ranging from −0.163 to 0.163 nm at intervals of 0.011 nm. For the fitting procedure between a total of 1271 simulated CBED patterns and one experimental CBED pattern, we use the χ2 value given by the following equation: (3) where Iith and Iiex are the theoretical and experimental intensities at the ith pixel, respectively. However, it is difficult to directly compare the entire CBED pattern from the raw images owing to the noise and the high-intensity distribution of the background. The background intensity distribution has to be subtracted by using the zero filtering method [11]. The mapping results of the χ2 values determined from the corresponding experimental images are displayed in Fig. 4f–i. The minimum point in each map is shown by ‘×’. In Fig. 4j–m, the best matching simulated CBED patterns are shown. In particular, the asymmetrical patterns of the (111) disks are well reproduced, where all the measured thicknesses were ∼130 nm. Therefore, it is confirmed that the patterns of the central disk change due to a crystal bending effect. From Fig. 4, it can be derived that the χ2 value is different for a compressive bend and for a tensile bend although the split HOLZ lines caused by a compressive bend or a tensile bend show similar splitting results. This result is explained by the fact that the passages of the electron beam in both the compressive bend and the tensile bend are handled as equivalent channels. Therefore, we can judge whether a crystal bending effect results in a tensile bend or a compressive bend by the differences in the values of the elastic and/or lattice constants between the matrix and the covered layer. In the present method, the dynamical effect plays an important role because our aim is to study the change in the CBED patterns formed by the ZOLZ reflections. In a kinematical simulation, only the projection of the channel of the incident electron beam is expressed. Therefore, it is not possible to discuss how an incident electron beam passes through the sample. On the other hand, the analysis of a dynamical simulation can help to easily determine whether the bend is a compressive bend or a tensile bend. Results of the measured bending angle of SiGe are displayed in Fig. 5. Almost all sample thicknesses measured by the present experiment are within a range from 100 to 200 nm. Here, errors induced in the present measurement are discussed. An error due to artificial measurement is considered to be fluctuations of the solutions of the pattern fitting procedure due to several factors such as the determination of the central point of an experimental CBED pattern, the estimation of background intensity and so on. These errors can be estimated by using the statistical average processing method. Five similar experimental patterns were taken from several regions. Then, each result obtained by the fitting procedure was averaged for the same observed region and the error was estimated as a function of the distance from the interface of Si/SiGe. In addition, the error due to the accelerating voltage determination is uniformly estimated to be θ = 0.003° by comparing the experimental image and the simulated results at the matrix region of Si. Therefore, the estimated total error is displayed by each error bar. Furthermore, it should be noted that the lower limitation of the present measurement is below 0.03°, from Fig. 5. This limitation is inferior to the measurement using split HOLZ lines. Fig. 5 Open in new tabDownload slide Measured bending angles as a function of the distance from the SiGe/Si interface. Fig. 5 Open in new tabDownload slide Measured bending angles as a function of the distance from the SiGe/Si interface. In this thickness range, the crystal bending effect caused by the thin-foil relaxation is almost independent of the sample thickness and rapidly decreases with the distance from the interface until a distance of 40 nm. It is noted that the bend effect can be measured even at the neighbourhood of the interface up to ∼5 nm. Because it is impossible to measure the bending angle in this region by using the split HOLZ lines, this fact is a significant advantage of the present method of measurement. The crystal bend of the present sample was a tensile bend, an extremely appropriate result obtained from the difference in the lattice constants between Si and SiGe. Concluding remarks It is found that a crystal bending effect is clearly included into the ZOLZ reflections of a CBED pattern. The bending effect can be measured easily by performing a pattern fitting between the experimental low-order zone-axis CBED patterns and the dynamical simulated patterns. The method has two advantages. One can easily judge whether the polarity of the crystal bend is compressive or tensile. Further, the crystal bending effect can be also measured at the sharp bending region where it is impossible to measure the crystal bending effect by using the split HOLZ lines. This research was partially supported by the Chiral Materials Research Center (Tokyo University of Science). 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TI - Quantitative and easy estimation of a crystal bending effect using low-order CBED patterns
JF - Journal of Electron Microscopy
DO - 10.1093/jmicro/dfn019
DA - 2008-12-01
UR - https://www.deepdyve.com/lp/oxford-university-press/quantitative-and-easy-estimation-of-a-crystal-bending-effect-using-low-cYVChrUh1D
SP - 181
EP - 187
VL - 57
IS - 6
DP - DeepDyve
ER -