Two-dimensional Gaussian fitting for precise measurement of lattice constant deviation from a selected-area diffraction map

Abstract

Unlike X-ray diffraction or Raman techniques, which suffer from low spatial resolution, transmission electron microscopy can be used to obtain strain maps of nanoscaled materials and devices. Convergent-beam electron diffraction (CBED) and nanobeam electron diffraction (NBED) techniques detect the deviation of a lattice constant (i.e. an indicator of strain) within 0.01%; however, their use is restricted to beam-insensitive samples. Selected-area electron diffraction (SAED) does not have such limitations but has low spatial resolution and precision. The use of a spherical aberration corrector and a nanosized selected-area aperture improves the spatial resolution, but the precision is still low. In this study, a two-dimensional stage-scanning system is used to acquire arrays of diffraction patterns at different positions of the sample under fixed beam conditions. Data processing with iterative nonlinear least-squares fitting enabled the spot displacement for each point of the scan area to be measured with precision comparable to that of the CBED or NBED technique. The precise strain determination, in combination with the simplicity of the measurement process, makes the nanosized SAED technique competitive with other methods for strain mapping at nanoscale dimensions.

selected-area diffraction, Levenberg–Marquardt method, two-dimensional Gaussian fitting, lattice constant measurements, two-dimensional sample scanning system, strain map

Introduction

The development of a method for nanoscale strain measurements has become an important challenge in recent years. Advances in electronics, where strained Si has become the basis of various technological processes [1], as well as the need to visualize strain to improve the performance of battery materials [2] necessitate the development of a reliable method for determining strain or lattice constant changes with high accuracy. Several different methods for accurately evaluating strain have been developed, e.g. X-ray diffraction (XRD) and Raman spectroscopy, which enable the detection of strain on the order of 0.01% [3]. However, the spatial resolutions of these methods are low [4,5], and their use is limited at the nanometre scale. As a powerful tool for observing the nanostructure of materials, transmission electron microscopy (TEM) provides a possible means of measuring strain at the nanoscale via electron diffraction techniques [6–8]. Convergent-beam electron diffraction (CBED) [9–11] and nanobeam electron diffraction (NBED) [12–14] have been used to accurately determine strain. A lattice constant change within 0.1% can be detected using these methods [9–14]. However, CBED requires relatively thick samples with a homogenous composition along the beam direction, and oriented away from low-index orientations such as <110> or <100>, [9] which is not always feasible. Moreover, the focused probe used in CBED may generate contamination, which can cause localized stresses [15]. The CBED and NBED techniques can also damage the sample through high electron doses to the evaluated area. This shortcoming is critical in the case of beam-sensitive battery materials.

Selected-area electron diffraction (SAED) can detect a deviation of the lattice constant in a local region of a material. However, because of deviation or drift of the magnetic lens and/or sample height, the lattice parameters determined by SAED typically vary by approximately several percent [16]. This uncertainty is much greater than that associated with XRD or CBED techniques. Another problem is the large area selection error due to spherical aberrations of the objective lens, which limits the spatial resolution of SAED to 100 nm. When a spherical aberration (C_s) corrector is used to reduce the C_s to almost zero, the area selection error can be drastically reduced and a smaller aperture can be used [17]. Recently, a nanosized SAED (NSAD) method for precise strain mapping was reported [18], where a two-dimensional (2D) sample scanning system [19] was used to acquire diffraction patterns at different points on the sample as a map. In this method, the optical system of a transmission electron microscope is fixed at certain conditions and is less susceptible to magnetic lens issues; therefore, a relative lattice constant change can be detected through measurement of the inter-spot distances at each point of the diffraction map. However, in the previous study, no attempt was made to maximize precision; more importantly, the precision was purposefully reduced to enable the acquisition of larger strain maps.

The aim of the present study is to improve and maximize the precision of the previously reported method by accurately determining the positions of diffraction spots and their displacement along the scan area. In this study, we do not determine the exact value of a lattice constant. We evaluate the changes of a lattice constant within the scan area by measuring the relative distance deviation between two symmetric diffraction spots. This relative distance deviation corresponds to the relative lattice constant changes within the scan area. Under typical experimental conditions, diffraction spots are spread over a few tens of pixels in the CCD detector; because of the noise, the highest pixel position does not always correspond to the centre of the spots. Therefore, measuring the distance between spots accurately without numerical processing is difficult. In this study, we applied the Levenberg–Marquardt method [20,21] to fit experimental diffraction spots with a 2D Gaussian function. The use of an iterative nonlinear least-squares fit helps to determine the diffraction spot position with subpixel accuracy; therefore, the precision of strain measurements by NSAD is substantially improved and is found to be comparable with the precision afforded by the CBED or NBED technique.

Methods

The experiments were performed in a C_s-corrected transmission electron microscope (JEOL JEM-ARM200F-G) operated at 200 kV (C_s = −52.94 nm) and equipped with a selected-area aperture with a diameter of 10 nm at the sample plane (Fig. 1); the aperture was fabricated using a focused ion beam.

Fig. 1.

Experimental setup. The double-tilt 2D stage-scanning system allows the diffraction map to be acquired under fixed beam conditions. A nanosized SA aperture in combination with a C_s corrector provides high-spatial resolution. The experimental diffraction spots were fitted with a 2D Gaussian function to determine the spot position with subpixel accuracy. OL, objective lens; DP, diffraction plane; IL, intermediate lens; PL, projector lens; SA, selected area.

The double-tilt 2D stage-scanning system [19] was used to adjust the sample orientation. The specimen position was controlled along two directions independently by two channels (x, y) powered by an amplified Gatan DigiScan II signal. The maximum displacements along the x and y axes (±0.19 μm and ±0.6 μm, respectively) were reached when ±40 V was applied to the piezoelectric actuators. For the chosen scan area of 400 × 400 nm² (x: ±40 V; y: ±14 V), the precisions of the displacements along the x, y directions are 0.3 nm and 0.25 nm, respectively. The high stability of the piezo-driven scanning system enabled us to move the sample within the scan area and simultaneously acquire diffraction patterns under fixed beam conditions. As a result, we obtained a four-dimensional (4D) dataset consisting of a 2D map of the diffraction patterns. Diffraction patterns were collected by a Gatan Orius-200D camera at a resolution of 2048 × 2048 pixels with dwell times between 8 and 16 s for data acquisition. The nominal camera length was set to 200 cm to use low-index diffraction spots, where the effect of distortion should be small [22,23]. The first intermediate and the condenser lens was carefully adjusted to minimize the size of the diffraction spots in the SAED pattern. Here, we note that subsidiary diffuse peaks with 4-fold symmetry often appear around the experimental spot because of the irregular shape of the nanosized SA aperture (Fig. 1). However, these peaks do not affect the Gaussian fitting results because the centre position of a peak is determined from half of the peak height. Therefore, the central position could be precisely determined even for a deformed peak if it had a symmetric shape.

Although deciding the absolute camera length is difficult, the relative change of a lattice constant can be precisely determined by measuring the inter-spot spacing of a diffraction pattern acquired in a 4D data set. Thus, the strain map can be readily drawn from the 4D data.

Conversely, if we assume the scan area to be single-phase and strain-free, the spread of the data expresses the precision of the method. Therefore, we plotted the distance deviation map that indicates the precision of the method by comparing the spot distance relative to that at the middle of the scan area. The relative displacement δ for each point of the scan was defined as δ = (d − d_ref) / d_ref × 100%, where d and d_ref are the distances between diffraction spots at selected and reference points, respectively. As a test sample for precision evaluation, we chose a Si (110) specimen thinned by the conventional Ar-ion milling technique. We assumed that the scan area was defect-free. Therefore, the distances between diffraction spots should be constants within the observed area. On the basis of this assumption, we expected that any deviation of spot distance could be attributed to the precision of the method. Experimentally obtained 2D spot profiles (Fig. 1) were fitted with a 2D Gaussian function using the Levenberg–Marquardt method [20,21] to determine the spot position with subpixel accuracy. This method has become the standard of nonlinear least-squares routines because it is very robust and stable method, demonstrating very high convergence at low signal-to-noise (S/N) ratio in comparison to other fitting methods [24,25]. We then measured the distance between two symmetric spots fitted with 2D Gaussians for each point of the scan area to plot the spot displacement maps to evaluate the precision.

Results and discussion

Beforehand, we defined the difference between accuracy and precision with reference to the strain measurement technique evaluation. Precision is the repeatability or reproducibility of the measurement. The precision of a strain measurement technique is defined by the standard deviation of the strain level within an unstrained reference area [26], which demonstrates the reliability of the measurement in terms of the smallest assessable variation of strain [13]. The accuracy is the degree to which the result of the strain measurement conforms to the actual strain value. In the current technique, because we measured the relative lattice constant changes, we mainly focus to the precision evaluation of NSAD.

To evaluate the precision of the proposed technique, we scanned a 400 × 400 nm² area (Fig. 2a), divided it into 25 points and measured diffraction patterns at every point of a 5 × 5 array (Fig. 2b). The displacement map of 111 spots indicates the changes within 0.18% with a standard deviation of 0.04% (Fig. 2c). These values correspond to the maximum data deviation and precision, respectively. We also evaluated the values for 222 spots, where the maximum deviation and precision were found to be 0.19% and 0.04%, respectively, as depicted in Fig. 2d. With the current camera length setting with a resolution of 2048 × 2048 pixels, the distance between 111 spots is ~862 pixels, whereas that between 222 spots is ~1727 pixels. Thus, the relative per-pixel displacement (the lattice constant change estimated by the one pixel shift) is 0.12% for 111 spots and 0.06% for 222 spots. The experimentally obtained precisions for both cases are better than the relative per-pixel displacement because of the 2D Gaussian fitting applied here. One may notice that displacement maps for 111 and 222 spots indicate some strains level but do not correlate to each other. The fluctuations on the maps originated by the experimental errors propagated to the results of Gaussian fitting. The level of these errors limits the precision and accuracy of proposed method.

Fig. 2.

(a) TEM image of the analysed area. The red line indicates the scan area. (b) Diffraction pattern corresponding to the centre of the scan area. The marked spots were used for strain evaluation; the insets show the shapes of selected spots. (c) The 111 spot displacement map. (d) The 222 spot displacement map. The coloured steps on the displacement maps correspond to the value of the standard deviation.

As evident in the map, the average strain level is not zero, even if the sample is claimed to be strain-free. This contradiction arises because we plot the map relative to the middle point of the scan area set as zero. In fact, this point is a randomly selected experimental point whose strain level has been artificially set to zero. Therefore, the average value of the strain differs from zero by the detected value of strains at the reference point.

Another set of data were acquired and analysed as shown in Fig. 3. In a manner similar to the first data set, diffraction patterns were acquired from a 400 × 400 nm² area divided into 25 points (Fig. 3a and b). In this case, the spot displacement map shows the changes within 3.4% with a standard deviation of 0.6% (Fig. 3c). If the observed area is assumed to be strain-free, these values correspond to the maximum deviation and precision, respectively. However, in comparison with the first data set, these values are very poor. Careful observation of the data reveals five potential outliers in the dataset. When these outliers are excluded from consideration (Fig. 3d), the precision improves by one order of magnitude (a maximum deviation of 0.21% with a standard deviation of 0.06%), although the obtained values are still worse than those for the first dataset. We next scanned the same area of the sample using a larger number of points (15 × 15) and obtained similar results: The distance between selected spots changed within 5% with a standard deviation of 0.7% (Fig. 3e). Statistical analysis again shows the presence of outliers in the same regions of the scanned area. In the absence of these outliers, the spread of the data became 0.53% with a standard deviation of 0.13% (Fig. 3f). The fact that the outlier positions are almost overlapped for the experiments with 5 × 5 and 15 × 15 points indicates that the observed outliers are not an accidental error.

Fig. 3.

(a) TEM image of the analysed area. The red lines delineate the scan area. (b) Diffraction pattern in the centre of the scan area. Marked spots used for strain evaluation. The insets show the shapes of selected spots. (c) Spot displacement map for the full-scan area (5 × 5 points). (d) Spot displacement map for the full-scan area without outliers (5 × 5 points). (e) Spot displacement map for row data (15 × 15 points). (f) Accuracy evaluation without outliers (15 × 15 points).

The colour step on the displacement maps corresponds to the value of the standard deviation.

To elucidate the phenomenon responsible for the outliers, we observed the 111 diffraction intensities on the corresponding points. As shown in Fig. 4, all of the statistical outliers are not only aligned and located in the same sites in the scan area but also show a strong correlation with the intensity of the selected diffraction spots. Upon overlapping the spot intensity and spot displacement maps, we find that all of the outliers correspond to a low intensity. At these points, S/N ratio becomes low; therefore, the Gaussian fitting is not reliable (Fig. 4a–d). This fact indicates the non-random nature of the deviation data.

Fig. 4.

Intensity maps and corresponding spot displacement maps: (a) $1\bar{1}1$ spot, 5 × 5 dataset; (b) $1\bar{1}1$ spot, 15 × 15 dataset; (c) $\bar{1}1\bar{1}$ spot, 5 × 5 dataset; (d) $\bar{1}1\bar{1}$ spot, 15 × 15 dataset and (e) $\bar{2}2\bar{2}$ spot, 15 × 15 dataset. Grayscale indicates the intensity value of the selected spot at each point of the scanned area.

We also noted that minimal intensity of the 111 spot corresponds to maximal intensity of the 222 spot (Fig. 4d and e). In the event of local bending of the sample, the surface curvature caused a local change in the Bragg condition: A strong Bragg condition is observed for 222 for some regions, causing the 222 spot intensity to increase and the 111 spot intensity to decrease. As a result, the S/N ratio for the 111 spot became low for these regions and the correct spot position could not be determined by 2D Gaussian fitting [24]. A similar situation may occur in the case of the outliers in the second dataset. Therefore, all outliers located in low-intensity areas must be omitted, as they appear only because of the error of Gaussian fitting and may not be related to the actual strain. This observation indicates the importance of the acquisition conditions, which should be free from changes in the Bragg condition, which may introduce a systematic error into the results. Other data obtained from the high-intensity points may correspond to the strain distribution within a bent area.

Despite the fact that the purpose of this work is to evaluate precision, we believe that demonstrating the sensitivity of the proposed technique is important. To this end, we chose a pulsed laser-deposited SrTiO₃ film on an SrTiO₃ substrate [27,28] as a test sample and prepared a slice using a focused ion beam. The lattice constant difference between the film and substrate was ~0.62% according to XRD analysis. We scanned the film/substrate interface and found that the average lattice constant difference between the film and substrate areas is approximately 0.60% (Fig. 5), which is in very good agreement with the results of the XRD analysis. However, an additional study is necessary to evaluate the highest sensitivity and improve the accuracy of the measurements.

Fig. 5.

(a) Bright-field cross-sectional TEM image of a PLD-grown STO film and (b) the corresponding strain map.

The effect of possible error sources (i.e. excitation error, distortion of diffraction pattern, inelastic scattering background) on the precision of our method is noteworthy. All of these parameters may lead to the displacement of diffraction spots from their actual position and hinder the determination of the actual value of the lattice constant. However, in this work, we evaluated the lattice constant changes within the scan area by measuring the relative deviation in distance between two symmetric diffraction spots. Therefore, we can disregard the effects of aforementioned error sources. For example, in the case of a 111 spot displaced 1% from its reference position by a change in the lattice constant, the error induced by the excitation error is only ~6 × 10⁻⁵%; for a 222 spot, the error is 2.5 × 10⁻⁴%. These values are negligibly small and do not affect the results. However, if our method is expanded to higher order reflections and/or tilted illumination conditions, noticeable displacements might be introduced because of excitation error. Similar to the effect of an inelastic scattering background and/or distortion, the relative change of a lattice constant can be accurately measured because the shift is small and because the inelastic background and/or distortion can be considered homogeneous.

In this work, we only used a pair of spots in a systematic row of diffraction spots to detect the deviation of the lattice constant along one crystal plane. However, simultaneous measurement of a network of diffraction spots would enable us to detect changes in the lattice constants in all directions simultaneously and with greater accuracy. We are therefore considering using such a method in future experiments.

Concluding remarks

SAED with a nanosized selected-area aperture enables the measurement of strain distribution with high-spatial resolution. Strain can be determined from the change in the inter-spot distances of diffraction spots, which is associated with deviations in the lattice constant. We applied the Levenberg–Marquardt method to fit the experimental diffraction spots with a 2D Gaussian function. Iterative nonlinear least-squares fitting was useful for determining the positions of diffraction spots with subpixel accuracy; the accuracy of strain measurements by NSAD were thereby substantially improved, becoming comparable to the accuracy of the CBED and NBED techniques. Analyses of strain-free areas indicated the possibility of determining the spot displacement with a precision of 0.04%. The precise strain determination in combination with potentially high sensitivity and simplicity of the measurement process makes the NSAD technique competitive with other methods for strain mapping at nanoscale dimensions. However, in the case of bent samples, the intensity of the selected spots must be examined closely, as Gaussian fitting is sensitive to the S/N ratio and may give incorrect results at low spot intensities.

Acknowledgements

The authors would like to thank Enago (www.enago.jp) for the English language review.

Funding

Part of this work was financially supported by the research project 'Advanced Low Carbon Technology Research and Development Program for Specially Promoted Research for Innovative Next Generation Batteries' of the Japan Science and Technology Agency (JST ALCA-SPRING) and by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) program for the Development of Environmental Technology using Nanotechnology.