Nanoinformatics pp 157-178 | Cite as

# Atomic-Scale Nanostructures by Advanced Electron Microscopy and Informatics

## Abstract

Interfaces dramatically affect the properties of materials because their atomic configurations often differ from the bulk material. A determination of the atomic structure of the interface is, therefore, one of the most significant tasks in materials research. Electron microscopy and theoretical calculations have been effectively used to accomplish this important task. In addition, an informatics approach has recently been combined with theoretical calculations to efficiently determine the atomic structures of interfaces. This chapter introduces the determination of interface structures using an informatics approach (Bayesian optimization and virtual screening) along with advanced electron microscopy. In the informatics approach, calculation acceleration on the order of 10^{6} can be achieved. Determination of the interface structure with resolution better than ~45 pm is now possible using advanced electron microscopy. In this way, nanostructures at grain boundaries and heterointerfaces can be qualified. We will introduce these state of the art methods to investigate nanostructures.

## Keywords

Interface Electron microscopy Scanning transmission electron microscopy Bayesian optimization Virtual screening## 8.1 Atomic Structures of Interfaces

Interfaces are a kind of lattice defect inside materials and can have significant effects on the overall material properties. For instance, interfaces in polycrystalline materials, i.e., grain boundaries (GB), determine the ion transportation properties and high temperature mechanical properties [1, 2, 3, 4]; an atomically controlled interface in a thin film often provides unique properties such as the formation of two-dimensional gases [5, 6]. The fact that interfaces have different properties from the bulk is a consequence of the fact that they have different atomic structures from the bulk. Thus, for a comprehensive understanding of interface properties, determination of the atomic structure of the interface is crucial.

Since the atomic structure of the interface is strongly dependent on the crystal orientation, lattice planes, and terminations, a systematic study of the interface structure is indispensable for achieving a comprehensive understanding. Thus, the atomic structures of interfaces have already been extensively investigated. Because these characteristic atomic configurations at the interface appear within a very limited area (below 10 nm), high spatial resolution observations using transmission electron microscopy (TEM) and theoretical calculations using atomistic simulations have been effectively applied to investigate interfaces.

Similar to TEM, aberration-corrected scanning transmission electron microscopy (STEM) has achieved sub-0.45 Å spatial resolution [7], and direct atom-by-atom imaging is now routinely possible via annular dark-field (ADF) imaging. In addition, owing to the rapid improvement in detectors, interface chemical analysis using energy-dispersive X-ray spectroscopy (EDS) and electron energy loss spectroscopy (EELS) can also achieve atomic resolution. In short, atomic-resolution STEM has become a very powerful tool for characterizing the atomic structures of interfaces [8, 9].

^{4}in even the simplified coincidence site lattice (CSL) grain boundary, namely Σ grain boundaries [10, 11]. In a straightforward manner, as schematically illustrated in Fig. 8.1, structure and energy calculations for all candidates must be performed, and leading to optimized configurations and energies of these are obtained (E

_{i,j}in Fig. 8.1). The most stable configuration with the minimal energy (E

_{i, mim}in Fig. 8.1) can then be determined from the density functional theory/molecular dynamics (DFT/MD) simulation of the interface. Furthermore, the same “brute force” computation is necessary to determine other types of interfaces because the interface structure is dependent on the type of interface (ΣGB

_{1}, ΣGB

_{2}, … ΣGB

_{N}in Fig. 8.1).

If the structure and energy of unknown interfaces could be determined more efficiently and accurately, the investigation of interfaces would be dramatically accelerated, which could lead to a deeper understanding of the mechanisms that give rise to interface properties. To more efficiently determine interface structures, a genetic algorithm method and a random structure searching algorithm method have been proposed [13, 14]. However, many trial calculations are still necessary to determine a single grain boundary structure. More recently, much more efficient methods based on machine learning techniques, including virtual screening and Bayesian optimization have been proposed by the present authors [12, 15, 16, 17]. Those methods are described below.

## 8.2 Informatics Approach for Interfaces

### 8.2.1 Virtual Screening

In this section, a virtual screening method for interface structure determination is described. Virtual screening is an effective method in time-critical problems and was applied to determine the structure and energy of an interface. This virtual screening technique has been used in drug discovery, where a prediction model was constructed using machine learning from a relatively small dataset and a large database consisting of the actual data and data predicted by the prediction model. Then, the candidate drug that is most likely to have the intended effectiveness is selected from the larger constructed database. More recently, this virtual screening method has been applied to discover new molecules for organic electro-luminescence (EL) applications and has succeeded in its discovery aims [18]. We have applied this virtual screening technique to predict the structure and energy of certain interfaces [12].

_{1}and ΣGB

_{2}. Once the predictor is constructed, the grain boundary energies can be predicted from the initial configurations. Then, the candidate configuration that is most likely to give the minimal energy E

_{i, mim (i=3, 4,… N)}can be determined. Next, the promising initial configuration is optimized using the structure and subsequent energy calculations. Finally, the accurate energy and stable structure are obtained (Stable ΣGB

_{3 ~ N}in Fig. 8.2).

_{1}and ΣGB

_{2}in Fig. 8.2. Those grain boundaries were selected as the training data based on the variance of their tilt angles and computational costs for their calculations. Structure and energy calculations for a total of 150,000 configurations, corresponding to approximately 15% of all possible configurations, were performed. We can confirm that the calculated structures are almost identical to the previously reported structures [19, 20], indicating that these training data are suitable for constructing the predictor.

List of descriptors

List of descriptors | |
---|---|

tan(θ/2) | Number of longer bond length |

sin(θ/2) | Average shorter bond length |

Atomic density around GB | Average longer bond length |

Average 1st NN (Near Neighbor) bond length | Shortest bond length |

Average 2nd NN bond length | Number of dangling bond around GB |

Average 1st NN bond length around GB | Relative translation distance along x direction |

Average 2nd NN bond length around GB | Relative translation distance along y direction |

Number of shorter bond length | Relative translation distance along z direction |

In addition to these descriptors, their square, inverse, exponential and exponential inverse values were considered. As a result, 83 descriptors were obtained, which were standardized to align their average and variance to zero and one, respectively.

The nonlinear support vector machine (SVM) method was used for regression analysis. In this study, the most stable structures and metastable structures of Σ5[001]/(210), Σ 5[001]/(310), Σ17[001]/(410), and Σ17[001]/(350) were considered for construction of the prediction model. We have selected those grain boundaries as the training data based on the variance of tilt angles and computational costs for their calculations.

There are two parameters in the SVM, the margin of tolerance and penalty factor. The best parameters were selected from combinations where the margin of tolerance was 0.001, 0.01, 0.05 or 0.1, the penalty factor was 10, 100, 1000 or 10000, and the variance was 10^{−2}, 10^{−3}, 10^{−4} or 10^{−5}, for a total of 64 different patterns. As a result, a margin of tolerance of 0.01, a penalty factor of 1000 and a variance of 10^{−4} were used as SVR parameters.

The results of the regression analysis for the training data are shown in Fig. 8.3a. Most data lie along the grey line, indicating that the predicted energies are equal to the accurate energies and that the regression analysis succeeded in correctly constructing the predictor. To evaluate the accuracy of the constructed predictor, the predictor was applied to Σ13[001]/(230) as a test situation. The results predicted by the predictor are shown in Fig. 8.3b. Most of the predicted grain boundary energies also lie on the grey line, indicating that the constructed predictor is also suitable for the test data. This result implies that the constructed predictor has the potential to predict the energy of the grain boundaries prior to the structure and energy calculations.

Here, we focus on the blue data point marked by the blue arrow in Fig. 8.3b. Based on the constructed predictor, the blue data point was predicted to provide the minimum grain boundary energy. It should be mentioned that the virtual screening method and the calculations of all candidates give the minimum grain boundary energy at the same blue data point. The predicted grain boundary energy is 0.96 J/m^{2}, which is only 10% larger than that the minimum grain boundary energy obtained from all-candidate calculations. It is also noteworthy that the predicted rigid body translation state (X = 5.0 Å, Y = 1.0 Å, and Z = 0.0 Å) is identical to the most stable rigid body translation state determined by all-candidate calculations.

We succeeded in screening all possible candidates and selecting the most promising candidate configuration for accurately provide the most stable structure. By performing the structure and energy calculation once for this rigid body translation state, a grain boundary energy and structure identical to those obtained from all-candidate calculations can be obtained. Namely, the stable grain boundary structure and energy can be determined with only a one-time calculation using the present virtual screening method, which is significantly more efficient than previously reported methods.

Since the constructed prediction model (the predictor shown in Fig. 8.2) was established, this predictor was also applied to other GBs. Here, based on the constructed predictor, the structures and energies of 12 other [001]-axis-symmetric tilt CSL grain boundaries, Σ25[001]/(430), Σ25[001]/(710), Σ29[001]/(520), Σ29[001]/(730), Σ37[001]/(610), Σ37[001]/(750), Σ41[001]/(910), Σ41[001]/(540), Σ53[001]/(720), Σ53[001]/(950), Σ61[001]/(11 1 0), and Σ125[001]/(11 2 0), are predicted.

### 8.2.2 Bayesian Optimization (Kriging) [15]

In this section, we demonstrate an alternative and powerful method that can be used to search for stable interface structures with the aid of a geostatistics approach called kriging. Kriging is an effective interpolation method based on a Bayesian optimization and Gaussian process governed by prior covariances. This Kriging method has been previously used to predict the optimum access points for geological mining operations. Here, we apply this Kriging technique to determine the stable structures of interfaces.

- (1)
Several initial configurations (20 configurations here) are randomly selected from the search space shown in Fig. 8.5a.

- (2)
The structure optimization and energy calculations for the selected configurations are performed, and a joint probability distribution is calculated with the Gaussian process considering the three rigid body translations along each direction as the descriptors. The grain boundary energy,

*E*_{ GB }, is then estimated. - (3)
Based on the joint probability distribution, the possible GB energies are estimated at all points in the search space.

- (4)Z-scores, calculated by the following equation, are estimated at all points in the search space.where$$ Z - score_{i} = (GB\,Energy_{current\,\hbox{min} } - GB\,Energy(x_{i} ))\,\text{/}\,\sqrt {\sigma (x_{i} )} $$
*GB Energy*_{current min}is the minimum GB energy at this moment, while*GB Energy(***x**_{ i }*)*and*σ(***x**_{ i }*)*are the mean and standard deviation at the point**x**_{i}in the search space respectively. - (5)
The point that has the maximum Z-score is selected as the next search point because that point is most likely to have the least GB energy at the moment.

- (6)
It is confirmed whether or not the acquired GB energies meet the convergence conditions, such as energy difference between the

*i*th calculation and the*i**+**1*th calculation.

The cycle of above operations ((2)–(6)) is repeated until the convergence criteria have been satisfied. In the structure optimization and energy calculation for (2), we have performed a static lattice calculation using an empirical potential method with the general utility lattice program (GULP) code [21]. The embedded atom potential method reported by Cleri et al. was employed [22].

^{2}. As can be seen here, the GB is composed of an array with a six-membered structure unit, in agreement with previously reported structures [23, 24]. However, 17,983 complete calculations were necessary to reach this stable structure using the conventional all-candidate calculation.

On the other hand, in the Kriging approach, the search space was interpolated based on the Gaussian process. We found that this Kriging approach greatly decreased the data necessary for calculations. In this case, the most stable point was determined after only 69 trials (including the initial 20 trials). The most stable structure obtained is shown in Fig. 8.6b; the GB is composed of a six-membered structure unit that is very similar to the stable structure obtained by comprehensive data searching. Furthermore, the calculated GB energy is 0.96 J/m^{2}, which is identical to that determined by the conventional method, indicating that our present method can accurately determine the most stable structure.

We repeated the Kriging operation 74 times for the same GB and found that the 43 Kriging operations were completed using fewer than 50 time calculations. As a result, the average number of calculations for determining the stable structures is 70. Based on this, the Kriging method has succeeded in accelerating the process of interface structure determination by ~150 times.

Since 17,466 configurations are present for this Σ3[110]/(111) GB bcc-Fe grain boundary, the Kriging method again achieves two orders of magnitude better efficiency than the conventional method. This clearly indicates that the Kriging method is a very powerful technique to determine the stable interface structure.

### 8.2.3 Kriging Method for Oxide Interfaces [16]

Comparing the virtual screening method and the Kriging method, the efficiency of the virtual screening is superior to that of the Kriging method. However, one has to construct a predictor in order to maintain this great efficiency. The most important advantage of the Kriging method is its wide applicability. No training is needed for the Kriging method, and thus it can be easily applied to other GBs in other materials. To show the wider applicability of the Kriging method, we used it to conduct similar studies on oxide interfaces. In particular, we applied the Kriging approach to grain boundaries of metal oxides including MgO, TiO_{2}, and CeO_{2} which commonly exhibit more complex structures than metals.

_{2}Σ5[001]/(210), and fluorite-CeO

_{2}Σ3[110]/(111) were selected to test the applicability of the present method. These grain boundaries have different complexities; the number of termination planes for MgO Σ5[001]/(210) and Σ5[001]/(310) is one (Fig. 8.9a, b), whereas that for TiO

_{2}Σ5[001]/(210) and CeO

_{2}Σ3[110]/(111) is two (Fig. 8.9c, d).

The same Kriging method was applied to these oxides GBs. Two hyper-parameters, pre-distribution and kernel parameter, were set to 0 and 3.0 respectively so that the kernel is not biased to 0 or 1. The random selection number for the initial calculation was set to 5, with the actual size of the three-dimensional rigid body translations in each xyz-direction acting as descriptors. Namely, smaller numbers than the above metal cases were used due to the higher computational cost of the oxide simulations.

For structure optimization and energy calculation, static lattice calculations with an empirical potential were performed using a general utility lattice program (GULP) code [21]. Buckingham-type potentials were used for MgO (Catlow et al. [27]), TiO_{2} (Bandura et al. [28]), and CeO_{2} (Minervini et al. [29]).

Regarding the convergence criteria in the Kriging method, the structure searching continues until five structures which have the identical lowest grain boundary energy are found. In this case, the grain boundary energies within ±0.005 J/m^{2} were judged to be the same. Until these convergence criteria are met, the Kriging algorithm continues searching for the lowest energy configuration.

We performed the Kriging method several times and confirmed that the number of calculations needed to reach convergence in these cases is 14–22, including the initial random sampling. This variation comes from the selection of the initial sampling. However, we would like to emphasize that the Kriging method is clearly powerful to search the most stable structure.

To confirm the applicability of this method to more complex structures, we applied it to TiO_{2} and CeO_{2} grain boundaries. The most stable structure of the rutile-TiO_{2} Σ5[001]/(210) grain boundary is shown in Fig. 8.10c. To maintain charge neutrality, the termination planes were set to Ti and O on their respective sides. A total of 21,630 structures and energies need to be computed in order to find the most stable structure using conventional brute force calculations, while the Kriging method can find the most stable structure by performing only 42 calculations, achieving a more than 500-fold efficiency. In addition, the obtained structure was compared with the previously reported one (Fig. 8.10c) [32], clearly showing that the structure determined by the present strategy was correct.

Next, the present method was applied to the fluorite-CeO_{2} Σ3[110](111) grain boundary. In contrast to the other three grain boundaries, this one possesses a [110] rotation axis. Notably, the Kriging method can determine the most stable structure (which is in agreement with the one reported previously [33]) using only 12 calculations. These results indicate that the Kriging method is applicable to complex oxides and can potentially achieve efficiency improvements by factors of ~10^{3}–10^{4} over the conventional all-candidate calculation method.

## 8.3 Microscopic Approach for Interfaces

### 8.3.1 Scanning Transmission Electron Microscopy (STEM)

STEM is one branch of TEM techniques that has been extensively used for characterizing interface structures in many materials and devices. In recent years, STEM combined with aberration correction technology has enabled direct atom-by-atom imaging via annular dark-field (ADF) imaging. ADF imaging uses a doughnut-shaped annular detector to selectively collect high-angle scattered electrons, building up images from the variation in this signal with probe position in a raster scan. Since the integrated intensity of high-angle scattered electrons strongly scales with the atomic number of the atoms under the probe, this imaging (so-called Z-contrast imaging) can sensitively visualize heavy element atoms [34]. However, ADF can seldom reliably visualize light elements due to their weak power to scatter electrons at higher angles. While ADF imaging mainly uses electrons scattered at high-angles to form atom images, there are many other possible detector geometries for collecting electron signals and forming images. One is known as annular bright-field (ABF) imaging, which involves the selective collection of electrons inside the bright-field disk via a small annular detector [35]. It has been shown that ABF imaging can directly visualize light atoms and can even directly visualize H atom columns inside compound materials [36, 37]. Since ADF and ABF images can be obtained simultaneously from the same sample positions, both heavy and light element atomic structures can now be directly visualized by combining these two images. However, it is still very difficult to identify the atomic species using only the ADF and ABF image contrasts, especially at interface regions where structure and chemistry are drastically changing. Since STEM uses an atomic-scale electron probe, STEM-based analytical techniques such as energy-dispersive X-ray spectroscopy (EDS) and electron energy loss spectroscopy (EELS) can also achieve atomic resolution. In particular, by utilizing ultrasensitive silicon drift detectors (SDDs) with much higher count rates, atomic-scale EDS mapping is now becoming possible [38]. This capability should be very powerful to directly characterize dopant/impurity segregation behaviors in grain boundaries and heterointerfaces. Thus, atomic-resolution STEM combined with spectroscopy will become an indispensable technique for characterizing atomic-scale structures and chemistry of interfaces.

### 8.3.2 Interface Structures Using Aberration-Corrected STEM

Two interface characterization studies using aberration-corrected STEM and EDS are highlighted in this section. One is on the solute segregation in a GB of ceramics [39] and the other is on the impurity segregation in a metal/ceramic heterointerface [40]. These studies demonstrate that atomic-resolution STEM is a powerful tool for directly understanding very complex segregation phenomena in materials.

#### 8.3.2.1 Solute Segregation Behavior of a ∑3 Grain Boundary in Yttria Stabilized Zirconia [39]

ZrO_{2} doped with Y_{2}O_{3} (YSZ) is one of the most important materials for use as an electrolyte in solid oxide fuel cells, where the overall ionic conductivity is strongly affected by the presence of GBs; such an effect may potentially originate from GB chemical inhomogeneity. Previous studies have shown that the Y solute atoms segregate to the GBs, and the amount of segregation is strongly dependent on the GB characteristics [41]. However, the atomic-scale mechanism of how Y solute actually segregate to GBs is still not well understood. In this study, we show an atomic-scale EDS mapping of a Ʃ3[110]/{111} model GB in YSZ.

It is noteworthy that in some atomic column layers, Y atoms are obviously depleted. This indicates that the Y atoms are not simply substituting in all of the cation sites around the GB to form segregation structures. Thus, we experimentally found that Y solute segregation formed atomically ordered extended structures across the GB within a range of approximately 3 nm. These experimental results are in good agreement with large-scale Monte Carlo simulations [39]. The simulation suggested that such processes can be driven by both the site-dependent segregation of Y due to strain and Y-*V* _{ O } interactions. Thus, recent advanced microscopy combined with theory can shed new light on the fundamental mechanism of solute segregation behaviors in GBs.

#### 8.3.2.2 Dopant Segregation Behavior in a Metal/Ceramic Interface [40]

Heterostructures between metals and ceramics have been widely used for power electronic devices requiring both high thermal performance and reliability in harsh environments. Since interfaces play a critical role in many properties, a fundamental understanding of the interface structure and formation mechanism is vitally important. One important possibility for obtaining heterointerfaces with better properties is to control dopant/impurity segregation behaviors. However, it has been very challenging to directly observe segregation structures at atomic dimensions in heterointerfaces. Here, atomic-resolution STEM-EDS mapping is shown to be a powerful tool for directly determining segregation structures in metal/ceramic heterointerfaces.

_{6}octahedron with rock salt structure. In the 2nd layer, a local maximum of Al can be found at the cation columns. Thus, the cation columns can be identified as Al columns. O and N could not be separated in the 2nd layer, but the ABF image contrast of the 2nd layer is similar to the Al-N contrast in the AlN bulk structure, although with inverted polarity (N-polar). Thus, we consider that the main structure of the 2nd layer is an AlN

_{4}tetrahedral monolayer. The polarity of this layer is inverted from the AlN substrate due to the presence of the MgO interlayer. Theoretical simulations suggested that the interface between Al metal and N-polar AlN is much more energetically stable than that between Al metal and Al-polar AlN.

Thus, Al alloy /AlN heterointerfaces should be stabilized by the formation of self-organized atomic-scale layered structures with Mg dopant segregation. Atomic-resolution STEM-EDS is a powerful tool for directly determining heterointerface structures with dopant/impurity segregation.

## Notes

### Acknowledgements

This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Nano Informatics” (Grant No. JP25106003) from the Japan Society for the Promotion of Science (JSPS).

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