# Periodic Nanostructures: 7 (Developments in Fullerene Science)

The k dependence indicates that Bayesian optimization functions are more efficient under more limited conditions. The absence of the m dependence indicates that efficiency is independent of N total , and thus, the comparable efficiency is expected in case of larger GNR systems. Such large structural sensitivity may have intriguing physics behind it; however, here, we focus on the general optimization problem. To gain quantitative insight into the efficiency of optimization process, we compare the optimization efficiency of with previously reported values. Note that, for the sake of comparison with the study of Ju et al.

Lower efficiency in the present study should occur because the optimal structure obtained by Ju et al. The successful results of optimizations for different properties with different models of the current study and the study of Ju et al.

The current study gives a rough indication for the efficiency of Bayesian optimization to be expected for other problems. As it is natural that optimization efficiency depends on the class of structures, the more important comparison here is the optimization efficiency of a single property versus that of multiple properties for the same system GNR. To this end, we compare the optimization efficiency of two different cases: The optimization for tends to be slightly faster than that for ZT.

Here, it is interesting that optimization for P , which is composed of S and R el , can be slower than that for ZT , as shown in fig. This is probably because ZT of model A is more sensitive to than to P , and the presence of in ZT makes the optimization process faster. To probe the relationship between geometrical structure and thermoelectric properties, we observe the optimized structures in detail. As for the P -optimized structure Fig. The structural optimization leads to strong flattening of electronic bands around energy levels of the edge state highlighted by bold lines in Fig.

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Nanostructuring in middle areas of zigzag GNRs, therefore, leads to phonon scattering without a significant change of the edge state and enhances the thermoelectric performance. Antidot GNRs model B , which will be used in the next section, are expected to benefit from this trend. As for the R th -optimized structure Fig. In such a structure, phonons are significantly localized and their group velocities are reduced, as seen in the longitudinal acoustic mode highlighted by bold lines in Fig.

The physical insights into the correlation between structure and thermoelectric properties described above can be fed back to formulate more efficient descriptors, although for the exploration of compound materials for specific physical properties good descriptors are somewhat nonintuitive For instance, Ju et al. We have also explored this possibility by using the insight that the labyrinthian shape tends to give high R th and adding the topological representation as a descriptor.

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As a result, an additional descriptor representing the labyrinthian shape was found to accelerate all the cases of the optimization for and most cases of the optimization for ZT. The details are described in section S3. Although the labyrinthian shape leads to highest ZT structures for model A, GNR structures with the edges remained should still be preferable because they are thermodynamically more stable and, moreover, available in experiments In this analysis, we define the structure obtained after calculating and processing structures with Bayesian optimization as the optimized one.

It is confirmed that the ZT is converged during the optimization process; the highest ZT does not change during the last calculations. Figure 3A compares thermoelectric properties— ZT , P , and R th —of the pristine structure black , the periodic antidot structure blue , and the optimal structure red , the last two of which are shown in Fig. The periodic antidot structure here means that antidots are introduced in all the 16 sections.

As shown in the figure, the optimal structure has an aperiodic array of antidots. These aperiodic optimal structures have also been obtained for superlattices Simply arranging the antidot periodically increases ZT 42 , in our case by 5. The current results show that optimization of the arrangement of antidots can effectively improve thermal and electrical properties simultaneously. This result indicates that the edge state is successfully used to independently tune thermal and electrical properties and that the potential of zigzag GNRs as thermoelectric materials is higher than that of armchair GNRs.

Note that, while the total number of sections 16 was chosen to be too large for an exhaustive search and reasonable size for the Bayesian optimization as discussed earlier, the optimal ZT increases with the number of sections because the structural degrees of freedom increase, which was also checked by performing the optimization for 8 sections.

The values are normalized by those of the pristine structure. B Periodic top and optimal aperiodic bottom structures. F LDOS distribution of the resonant states of the periodic top and optimal bottom structures. The resonant numbers correspond to those in E. To gain insights into the improvement of the thermoelectric performance of zigzag GNRs, we investigate their phonon and electron transport properties in detail.

In periodic structures, approaches the Bloch limit, , with increasing the number of antidots N dot because of the generation of Bloch states in the nanostructured region The dependence of on N dot shown in fig. For the modulation of , mainly the following factors compete with each other: The former effect becomes stronger with increasing number of antidots, while the latter effect becomes stronger with increasing length of the homogeneous region, that is, the region with a sequence of the same type of sections with or without an antidot.

Therefore, it is reasonable to suppose that the spatial distribution of smaller and larger antidots-distance provides tunability of the balance between the above two effects, although this physical knowledge alone does not explain why this particular distribution is optimal; thus, machine learning was needed to find the optimal structure. Unlike pristine GNRs that have a low Seebeck coefficient due to the absence of the band gap, periodic antidot GNRs have a higher Seebeck coefficient because of the presence of transport gaps energy gaps in electron transmission , corresponding to energy gaps of an infinite periodic antidot GNR.

The introduction of periodic antidots, therefore, can enhance the thermoelectric performance Fig. Utilization of the edge state requires suppression of the electron transmission of resonant states with energy near that of the edge state, as discussed in section S6. In general, any electron states are localized, and their transmission is suppressed in infinite random potential fields 41 , The indices of resonant state in Fig.

LDOS in the finite periodic structure spreads over the whole nanostructured region, while the optimal aperiodic structure leads to the localization of states in limited areas. It is intuitively comprehensible that the widely spreading states, which can be regarded as Bloch states, contribute to electron transport, while the strong localization generates the region with extremely low DOS and suppresses the electron transport, as shown in Fig.

The above discussions raise the question: How important is the exact arrangement of antidots to control thermoelectric properties? To answer this question, we calculate the distribution of thermoelectric properties for structures with the same numbers of antidots and homogeneous regions.

Since the number of homogeneous regions is an indication of their lengths, following the above discussion, the two numbers represent i surface scattering effect and ii interference effect. The optimal structure has 10 antidots and 7 homogeneous regions Fig. Figure 4 shows the distributions of P blue , gray , and R th red of structures with the same numbers as that of the optimal structure.

These three properties are normalized by those of the pristine GNR, and the data are extracted from structures calculated during the optimization process. The result shows that, on average, R th increases by 3.

In addition, exact replication of the obtained optimal arrangement of antidots in the experiment may not be required to increase R th because of their narrow distribution; the sample SD of R th is 0. Conversely, the wide distribution of P makes the arrangement of antidots more important; the SD of P is 0.

While P can increase by 3. The above discussion strongly suggests that the control of nanostructures is required for enhancing electron transport and thus for thermoelectric performance.

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## Fullerene chemistry

As for the overall uniqueness of the optima, there are other structures that exhibit ZT inferior to the optimal structure but not by much; as shown in fig. These structures have narrow ranges of numbers of antidots 8 to 12 and homogeneous regions 5 to 7. This result shows that these numbers can be a good indicator function for high- ZT structures, while we still need the structural optimization because of the fluctuation of thermoelectric properties within structures having the same numbers.

It is worth noting that such uniqueness of the optima varies for different models and problems. P blue , gray , and R th red of structures with the same numbers of antidots and homogeneous regions as the optimal structure. The thermoelectric properties are normalized by the value for the pristine GNR. The fitting curves of Gaussian distributions are shown as a reference. In summary, we have shown that the Bayesian optimization technique accelerates structural optimization including multitransport properties, namely, R th , R el , and S.

The analysis with the periodically nanostructured GNR reveals the following design strategies for thermoelectric nanostructures: Using the former strategy, we show that introduction of antidots can significantly enhance the thermoelectric performance of zigzag GNRs up to 11 1. Conversely, the careful arrangement of antidots is indispensable for keeping optimized electron states in the nanostructured region; otherwise, the electron properties fluctuate widely.

While the combination of these techniques may still be challenging at present, the fabrication of GNR structures proposed in this study should become possible in the near future. Furthermore, we may be able to fabricate arrayed antidot GNR structures combining nanomesh structures 56 with the above fabrication techniques; because neck regions in nanomesh structures can be regarded as arrayed GNRs, they will become arrayed antidot GNRs if antidots are introduced into the neck regions.

The demonstrated framework enables us to optimize structures without any previous knowledge of the relation between the structures and properties, and with intuitive descriptors to represent the structures.

### INTRODUCTION

The Bayesian optimization method is expected to be useful for searches of other geometries and physical properties as long as the number of candidates is processable and the properties are calculable. Before calculation of thermoelectric properties, a relaxation calculation was performed until the maximum atomic force became less than 0. For model B, to perform more realistic calculations, the effects of terminal hydrogen atoms were treated explicitly.

The optimized Tersoff potential was applied to C—C interatomic interactions, and the Brenner potential 62 modified for different properties including thermal properties 63 was applied to C—H and H—H interatomic interactions. Bayesian optimization has an advantage over empirical optimization in its ability to determine required data and a regression curve by machine learning without any previous knowledge of the model. Bayesian optimization was performed by repeating the following process: The Bayesian linear model with the random feature map approximates the Gaussian process, which has merit in the flexible expressivity by nonparametric regression.

The Gaussian process was widely used for materials design with the Bayesian optimization 11 , The next point for the exact calculation was determined following Thompson sampling More detailed explanations of the Bayesian optimization method can be found in the studies of Ju et al. Supplementary material for this article is available at http: Acceleration of the Bayesian optimization using a topological descriptor: The mean shortest path. References 71 — This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license , which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address. Science Advances 15 Jun Efficient multifunctional materials informatics enables the design of optimal graphene thermoelectrics. Abstract Materials development often confronts a dilemma as it needs to satisfy multifunctional, often conflicting, demands. Download high-res image Open in new tab Download Powerpoint.

Bayesian-based multifunctional structural optimization Before discussing the optimization, we first highlight the sensitivity of ZT to GNR structures. Observation of optimal structures To probe the relationship between geometrical structure and thermoelectric properties, we observe the optimized structures in detail. Optimization of antidot GNR structures Although the labyrinthian shape leads to highest ZT structures for model A, GNR structures with the edges remained should still be preferable because they are thermodynamically more stable and, moreover, available in experiments Impact of aperiodicity The above discussions raise the question: METHODS Analysis method Before calculation of thermoelectric properties, a relaxation calculation was performed until the maximum atomic force became less than 0.

Bayesian optimization Bayesian optimization has an advantage over empirical optimization in its ability to determine required data and a regression curve by machine learning without any previous knowledge of the model. Efficiency of single-functional optimizations section S3. Acceleration of the Bayesian optimization section S4. Structural optimization of antidot armchair GNRs section S5. Phonon and electron transport in finite periodic structures section S6. Effects of resonant states on thermoelectric properties section S7.

Resonant states in one-dimensional tight-binding chains section S8. Statistical errors of the efficiency of the Bayesian optimization section S9. Uniqueness of the optimal structures fig. Comparison of optimization efficiency of different functionals. Phonon thermal resistance in finite periodic structures. Electron transport in finite periodic antidot structures.

Effects of resonant states on thermoelectric properties. Electron transport properties of one-dimensional tight-binding chains. Efficiency of the Bayesian optimization. Rajan , Materials informatics. Today 8 , 38 — 45 Tanaka , Accelerated discovery of cathode materials with prolonged cycle life for lithium-ion battery.

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Koebel , Mechanical properties of monolithic silica aerogels made from polyethoxydisiloxanes. Narumanchi , Nanothermal interface materials: Technology review and recent results. We use cookies to help provide and enhance our service and tailor content and ads. By continuing you agree to the use of cookies. Arabian Journal of Chemistry Available online 18 May Under a Creative Commons license. Abstract This review is provided a detailed overview of the synthesis, properties and applications of nanoparticles NPs exist in different forms.

Graphical abstract Download high-res image 82KB Download full-size image. FESEM field emission scanning electron microscopy. TEM transmittance electron microscopy. XPS X-ray photon spectroscopy.

## ENTROPY DUE TO FRAGMENTATION OF DENDRIMERS

DRS Diffuse reflectance spectroscopy. FT-IR Fourier transform infrared. SERS surface enhanced raman spectroscopy. MA-SiO 2 methacrylate-functionalized silica. TMD-NDs transition-metal dichalcogenide nanodots. LSPR localized surface plasmon resonance. CVD chemical vapor deposition.