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CONDENSED MATTER PHYS I CS

1Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo,Bunkyo 113-8654, Japan. 2National Institute for Materials Science, 1-2-1 Sengen,Tsukuba 305-0047, Japan. 3Japan Science and Technology Agency, Core Researchfor Evolutional Science and Technology, 4-1-8, Kawaguchi, Saitama 332-0012, Japan.4RIKEN Center for Advanced Intelligence Project, 1-4-1 Nihonbashi Chuo-ku, Tokyo103-0027, Japan.*Corresponding author. Email: shiomi@photon.t.u-tokyo.ac.jp

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Multifunctional structural design of graphenethermoelectrics by Bayesian optimizationMasaki Yamawaki1, Masato Ohnishi1, Shenghong Ju1,2, Junichiro Shiomi1,2,3,4*

Materials development often confronts a dilemma as it needs to satisfy multifunctional, often conflicting, demands.For example, thermoelectric conversion requires high electrical conductivity, a high Seebeck coefficient, and lowthermal conductivity, despite the fact that these three properties are normally closely correlated. Nanostructuringtechniques have been shown to break the correlations to some extent; however, optimal design has been a majorchallenge due to the extraordinarily large degrees of freedom in the structures. By taking graphene nanoribbons(GNRs) as a representative thermoelectric material, we carried out structural optimization by alternating multi-functional (phonon and electron) transport calculations and Bayesian optimization to resolve the trade-off. As aresult, we have achieved multifunctional structural optimization with an efficiency more than five times thatachieved by random search. The obtained GNRs with optimized antidots significantly enhance the thermoelectricfigure of merit by up to 11 times that of the pristine GNR. Knowledge of the optimal structure further provides newphysical insights that independent tuning of electron and phonon transport properties can be realized by makinguse of zigzag edge states and aperiodic nanostructuring. The demonstrated optimization framework is also usefulfor other multifunctional problems in various applications.

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INTRODUCTIONMaterials informatics (MI) is an emerging approach that aims to accel-erate materials development by combining material science and datascience (1). The core of the approach is to identify “best” materials bybridging the gap between the data in hand and the optimal structureand/or composition using unrecognized complex correlation in thedata instead of (or in addition to) relying on insights driven by physical/chemical principles. MI has been applied to the design of novel ma-terials such as cathode materials for lithium-ion batteries (2), nitridesemiconductors composed of earth-abundant materials (3), piezo-electric materials (4), and thermoelectric materials (5–9). All of thesestudies have aimed to realize high-throughput screening of the bestmaterials from the pool of stoichiometric compounds.

Another but more ambitious course of MI aims to create nano-structures by identifying optimal geometry that maximizes the tar-get property. There has been research on nanoparticles embedded inmatrix to modulate heat conduction (10), solid-solid interfaces toidentify energetically stable structures (11, 12), and multicore struc-tures of plasmonic nanowires to control optical scattering and cloak-ing effects (13). Recently, the method has been extended for designingatomic-scale structures with minimum/maximum lattice thermal con-ductance (14, 15). There, to reduce the number of candidates for whichproperty calculations are performed, atomic-scale phonon transportcalculations and an informatics-based optimization technique are alter-nately conducted. The work has shown that such an approach canconsiderably accelerate nanostructure design for transport properties,which is not only limited to phonons but also applicable to any otherquasi-particles.

Now, the next nontrivial challenge is to extend such an approachto optimize the materials for multifunctional properties. In many ap-

plications, a material needs to satisfy multifunctional demands that areoften in trade-off relations, such as thermal resistance versus me-chanical strength in thermal insulators (16), and thermal conductanceversus mechanical compliance in thermal interface materials (17). Anextreme example in this context is thermoelectric materials, where op-timization requires resolution of the conflicting demands of Seebeckcoefficient, electrical conductivity, and thermal conductivity. Here, thedimensionless figure of merit is expressed as ZT = RthS

2T/Rel, whereRel/th is the electronic/thermal resistance, S is the Seebeck coefficient,and T is the temperature.

Thermoelectric materials have been widely studied, particularlywith increasing demands to power ubiquitous sensors and transmit-ters without a complex system and frequent maintenance for thecoming trillion sensor era (18). This has led to the enhancementof ZT particularly by using nanostructuring techniques, such as inlow-dimensional materials (19), superlattices (20, 21), nanocrystals ornanocomposites (22–24), and nanoporous structures (25, 26). However,the development of nanostructures requires much time and effort be-cause of its huge parameter space, much larger than that of single-crystalcompounds; thus, further acceleration of the development process requiresa goal, that is, knowledge of optimal structures, to guide the research.

Here, structural optimization is carried out by alternating multi-functional (phonon and electron) transport calculations and Bayesianoptimization. Bayesian optimization should be particularly useful foroptimizing thermoelectric structures, as the function for ZT is highlyuncertain because of the complex correlations among nanostructures,thermal properties, and electrical properties, and evaluation of ZT thatrequires calculations of both thermal and electrical properties is ex-pensive. We take graphene nanoribbons (GNRs) as the representativethermoelectric materials. Carbon nanomaterials, including GNRs, arepromising thermoelectric materials (27–30) due to high Seebeck co-efficient (31), high electron mobility (32), and mechanical flexibility(33–35), although their thermal conductivity is high (36, 37). To en-hance thermoelectric properties of GNRs, different types of nanostruc-tures have been proposed, such as vacancies (38, 39), edge disorders(40, 41), nanoporous structures (42, 43), chevron-type GNRs (44), andGNRs composed of different edges and widths (44, 45). It is also

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known that GNRs have different electronic properties depending ontheir edge type: armchair or zigzag edge. Armchair GNRs with a well-defined width are intrinsically semiconducting and thus have a highSeebeck coefficient (up to a few millivolts per kelvin) (46, 47). On theother hand, because of the edge state, which is a topologically stableelectronic state (48, 49), pristine zigzag GNRs are metallic and thushave a low Seebeck coefficient regardless of their width (30). We usezigzag GNRs in the present study, expecting that the presence of theedge state gives larger tunability of electrical and thermal transport.

This paper is organized as follows. In the first part, we demonstratethe effectiveness of the optimization method for thermoelectric nano-structures using periodically nanostructured GNRs. In the second part,we optimize the structure of antidot GNRs, which have recently beenfabricated in experiments using the ion beam technique (50, 51), andshow how the optimization helps identify high figure-of-merit struc-tures and understand pathways to separately control electron and pho-non transport.

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RESULTS AND DISCUSSIONNanostructured thermoelectric GNRsWe consider two different models as shown in Fig. 1: periodicallynanostructured GNR (model A) and antidot GNR (model B). A pris-tine GNR structure is defined by the indices (Nx, Ny), where Nx and Ny

are the numbers of carbon chains along the axial direction and thedirection perpendicular to the axis, respectively. Model A (Fig. 1A)is composed of a periodic structure of a (4, 6) GNR from which matoms are removed. In this analysis, we assume that the structural sta-bility is ensured if every carbon atom in the GNR structure belongs toat least one complete hexagonal lattice and no hexagonal lattice isisolated. In the case of model A, to validate the effectiveness of theBayesian technique for structural optimization of multiple transportproperties, the system is deliberately set small enough that we are ableto calculate ZT of all the candidates. While thermoelectric propertiesvary with the chemical potential (m), the representative values of ther-moelectric properties were evaluated at the “peak chemical potential”(mpeak) that gives maximum ZT within a realistic chemical potential

Yamawaki et al., Sci. Adv. 2018;4 : eaar4192 15 June 2018

range, −1.0 ≤ m ≤ 1.0 eV. To perform optimization based on machinelearning, the representations identifying each candidate, called “de-scriptors” or “features,” play an important role, although it is still con-troversial how to decide them (52). In this analysis, we use the binary flagdescribing the state of each hexagonal lattice as a descriptor: “1” and“0” for the complete and defective hexagonal lattice, respectively, asshown in Fig. 1A. The set of the binary flag descriptors for each struc-ture is defined as gA = {1, 0, …}.

For a system that is more realizable in experiments, we considerantidot structures, model B, shown in Fig. 1B. Model B is composed ofan antidot nanostructured region with multiple sections of (6, 8) GNRconnected with semi-infinite pristine GNRs with a width of Ny = 8.Each section in the nanostructured region is either a pristine structureor an antidot structure with a pore (diameter d = 2.9 Å) in the centerof the section. The binary flag descriptor for model B is defined to be 0and 1 for the pristine structure and the antidot structure, respectively.Similar to the case of model A, the set of the binary flag descriptors formodel B is defined as gB = {1, 1, …}. For model B, we mainly studysystems with 16 sections that give 32,896 candidates, which is so largethat the direct simulations of all the candidates would exceed the com-putational resource, but reasonably large for the Bayesian optimization.

Bayesian-based multifunctional structural optimizationBefore discussing the optimization, we first highlight the sensitivityof ZT to GNR structures. Figure 2A shows ZaveT (circle), ZmaxT(triangle), and ZminT (inverse triangle) of model A with differentnumbers of vacancies (m = 0 to 10), where ZaveT, ZmaxT, and ZminTdenote average, maximum, and minimum ZT, respectively, of all thecandidates with the same number of vacancies (m). While ZT of thepristine GNR with no vacancies (m = 0), Z0T, is relatively small (Z0T =0.048), introduction of vacancies enhances ZT on average. Note herethat vacancies do not always enhance ZT but can degrade ZT from thatof the pristine case. The large difference between ZmaxT and Z0T orZaveT shows that the thermoelectric performance of GNR is highly sen-sitive to the structure, and thus, it is an appropriate system to targetwith the optimization technique. Here, we consider not only the abso-lute ZT value but also its increment against m, because improving ZT

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Fig. 1. Nanostructured thermoelectric GNRs. (A) Periodically nanostructured GNR (model A) and (B) antidot GNR (model B). The size of the GNR section isdetermined by the index (Nx, Ny), where Nx and Ny are the numbers of carbon chains along the axial direction and the direction perpendicular to the axis, respectively.Binary flags represent the structural descriptor; 0 and 1, represent, respectively, defective and complete hexagonal lattices for model A and the pristine and antidot GNRsections for model B.

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with fewer vacancies is beneficial in terms of mechanical stability. Here,the average rate of increase in ZT with respect to m, namely, the gra-dient of dashed lines in Fig. 2A, is defined as D(ZT)/Dm. The beststructure also stands out in this context as, for example, D(ZmaxT)/Dm = 0.086 is three times higher than D(ZaveT)/Dm = 0.029 in therange of 1 ≤ m ≤ 10.

The multifunctional structural optimization for model A is per-formed with the Bayesian optimization, and its efficiency is comparedwith random search for different m (m = 6 to 10). For each m, NBayes

and N rand are evaluated, where NBayes/rand denotes the average numberof calculations needed until, for the first time, a structure that belongsto the top k% (k = 0.0 to 2.0%) of all the candidates is found. For theBayesian search, the values for 100 optimization runs are averaged,while, for the random search, Nrand is obtained with the hyper-geometric distribution: Nrand = (Ntotal + 1)/(kNtotal + 1), with Ntotal be-ing the number of all the candidates. Note that we omit the structureswith translational and reversal symmetries from the candidates to savethe optimization time.

The Bayesian search accelerates the exploration of high-ZT struc-tures (NBayes/Nrand < 1) in all cases as shown in Fig. 2B, except for asingular point for the best case ofm = 6. In most cases, the top 0.5% ofthe structures can be found by the Bayesian search with half the cal-culations for the random search (NBayes/Nrand < 0.5), and in the bestcase, which corresponds to the search of the best structure for m = 7,the Bayesian search requires merely 18% calculations of that with therandom search (NBayes/Nrand = 0.18). As for the dependence of effi-

Yamawaki et al., Sci. Adv. 2018;4 : eaar4192 15 June 2018

ciency on k andm, the efficiency of the Bayesian search increases withdecreasing k, while we do not observe a clear dependence on m. The kdependence indicates that Bayesian optimization functions are moreefficient under more limited conditions. The absence of the mdependence indicates that efficiency is independent of Ntotal, and thus,the comparable efficiency is expected in case of larger GNR systems.We note that the singular case form = 6 is due to strong dependenceof electronic properties on atomic geometries; low ZT structures thatare similar (closer in descriptor space) to the optimal structure makethe prediction of high-ZT structures difficult. Such large structuralsensitivity may have intriguing physics behind it; however, here, wefocus on the general optimization problem.

To gain quantitative insight into the efficiency of optimizationprocess, we compare the optimization efficiency of Rph

th with previouslyreported values. For Si/Ge interfacial structures, Ju et al. (14) obtained3.8 times higher efficiency (NBayes/Ntotal = 0.0089) than the presentstudy (NBayes/Ntotal = 0.034). Note that, for the sake of comparisonwith the study of Ju et al. (14), we performed additional optimizationcalculations without considering the reversal symmetry and usedNBayes/Ntotal as a criterion, where NBayes/Ntotal is obtained by averagingresults of 10 and 100 optimization runs for the study of Ju et al. (14)and this study, respectively. Lower efficiency in the present study shouldoccur because the optimal structure obtained by Ju et al. (14) is simpler,namely, a superlattice with smooth interfaces. The successful resultsof optimizations for different properties with different models of thecurrent study and the study of Ju et al. (14) indicate that Bayesian

Fig. 2. Multifunctional structural optimization using model A. (A) Maximum (triangle), minimum (inverse triangle), and average (circle) ZT of all the candidates form = 0to 10. Dashed lines show the linear fitting in 1 ≤ m ≤ 10. (B) Efficiency of the Bayesian optimization. NBayes/rand is the average number of calculations needed until, for the firsttime, a structure that belongs to the top k% of all the candidates is found with the Bayesian/random search. (C and D) Optimal geometrical structure for P/Rth and its electron/phonon band structure with that of the pristine GNR. The P- and Rth-optimal structures are GNRs with m = 4 and 7, respectively. Bold lines in the electron and phonon bandstructures show the edge state and the longitudinal acoustic band, respectively. For the electron band structure, the Fermi level is set to zero.

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optimization is versatile for various structural optimization problems.The current study gives a rough indication for the efficiency of Bayesianoptimization to be expected for other problems.

As it is natural that optimization efficiency depends on the class ofstructures, the more important comparison here is the optimization ef-ficiency of a single property versus that of multiple properties for thesame system (GNR). To this end, we compare the optimization effi-ciency of two different cases: optimization forRph

th (single-function) andthat of ZT (multifunction). The optimization for Rph

th tends to be slightlyfaster than that for ZT. Here, it is interesting that optimization for P,which is composed of S andRel, can be slower than that forZT, as shownin fig. S1. This is probably because ZT of model A is more sensitive toRphth than to P, and the presence of Rph

th in ZT makes the optimizationprocess faster. Form = 7, optimal structure with maximum ZT has thehighest Rph

th among all the candidates.

Observation of optimal structuresTo probe the relationship between geometrical structure and thermo-electric properties, we observe the optimized structures in detail. Figure 2(C and D) shows GNR structures with m = 4 and 10 that have thelargest power factor P (= S2/Rel) and Rth, respectively, in all the can-didates for m = 1 to 10. As for the P-optimized structure (Fig. 2C),vacancies are introduced over the entire area of GNR, except for thehexagonal lattices along the edge. The structural optimization leads tostrong flattening of electronic bands around energy levels of the edgestate (highlighted by bold lines in Fig. 2C), and eventually, band gapsare generated. Nanostructuring in middle areas of zigzag GNRs, there-fore, leads to phonon scattering without a significant change of theedge state and enhances the thermoelectric performance. AntidotGNRs (model B), which will be used in the next section, are expectedto benefit from this trend. As for the Rth-optimized structure (Fig. 2D),which also gives the highest ZT, the GNR has a complicated labyrin-thian shape. In such a structure, phonons are significantly localizedand their group velocities are reduced, as seen in the longitudinalacoustic mode highlighted by bold lines in Fig. 2D, for example.

The physical insights into the correlation between structure andthermoelectric properties described above can be fed back to formulatemore efficient descriptors, although for the exploration of compoundmaterials for specific physical properties good descriptors are somewhatnonintuitive (52). For instance, Ju et al. (14) reduced the number of tar-get structures by using the knowledge learned from precalculations ofsmaller systems. We have also explored this possibility by using theinsight that the labyrinthian shape tends to give high Rth and addingthe topological representation as a descriptor. As a result, an additionaldescriptor representing the labyrinthian shape was found to accelerateall the cases of the optimization for Rph

th and most cases of the optimi-zation for ZT. The details are described in section S3.

Optimization of antidot GNR structuresAlthough the labyrinthian shape leads to highest ZT structures formodel A, GNR structures with the edges remained should still be pref-erable because they are thermodynamically more stable and, more-over, available in experiments (50). We therefore apply Bayesianoptimization to search for the antidot GNR (model B) with highestZT. In this analysis, we define the structure obtained after calculatingand processing 3000 structures with Bayesian optimization as the op-timized one. It is confirmed that the ZT is converged during the op-timization process; the highest ZT does not change during the last2469 calculations. Figure 3A compares thermoelectric properties—

Yamawaki et al., Sci. Adv. 2018;4 : eaar4192 15 June 2018

ZT, P, and Rth—of the pristine structure (black), the periodic antidotstructure (blue), and the optimal structure (red), the last two of whichare shown in Fig. 3B. The periodic antidot structure here means thatantidots are introduced in all the 16 sections. As shown in the figure,the optimal structure has an aperiodic array of antidots. These aperi-odic optimal structures have also been obtained for superlattices (14).The optimal arrangement of antidots increases ZT by 11 times (fromZT0 = 0.044 to ZT = 0.46), with the increase in P and Rth by 2.9 and3.6 times, respectively. Simply arranging the antidot periodically in-creases ZT (42), in our case by 5.0 times, compared with the pristinestructure, yet the remaining 2.1 times does require the optimization.Note that although the ZT of a periodic antidot GNR was calculated toreach 1.2 in the study of Yan et al. (42) using a simple tight-bindingapproximation with the nearest-neighbor p-orbital and without struc-tural relaxation, when calculating the same structure using the currentmethod with transfer integrals of 1s orbital of hydrogen and 2s2px/y/zorbitals of carbon with a longer cutoff length (= 3.7 Å) and structuralrelaxation, we obtain ZT = 0.28. The current results show that opti-mization of the arrangement of antidots can effectively improvethermal and electrical properties simultaneously. Furthermore, com-pared with additionally performed analyses for armchair GNRs witha similar width (Ny = 6), ZT of the optimal zigzag GNR is 1.2 (3.3)times as high as that of the optimal (pristine) armchair GNR (see sec-tion S4). This result indicates that the edge state is successfully used toindependently tune thermal and electrical properties and that thepotential of zigzag GNRs as thermoelectric materials is higher thanthat of armchair GNRs. Note that, while the total number of sections(16) was chosen to be too large for an exhaustive search and reason-able size for the Bayesian optimization as discussed earlier, the optimalZT increases with the number of sections because the structuraldegrees of freedom increase, which was also checked by performingthe optimization for 8 sections.

To gain insights into the improvement of the thermoelectric per-formance of zigzag GNRs, we investigate their phonon and electrontransport properties in detail. Figure 3C shows the phonon transmissionfunctions Qph(w) of the pristine (black), periodic (blue), and optimalaperiodic (red) structures. Qph(w) decreases, and hence Rph

th increases,significantly by introducing the periodic antidots. In periodic struc-tures,Rph

th approaches the Bloch limit,Rph;∞th , with increasing the num-

ber of antidots (Ndot) because of the generation of Bloch states in thenanostructured region (53). The dependence ofRph

th onNdot shown infig. S4 indicates thatRph

th of the periodic structure withNdot = 16 reaches99% ofRph;∞

th (= 0.86 K/nW). Here, it is worth mentioning that Rphth ap-

proachesRph;∞th more rapidly than previous study on Si/Ge superlattice,

which suggested that thermal resistance was proportional to the inverseof the number of periods (53). Optimization can further increase Rph

thwith the decrease of Qph(w) over a wide energy range despite the in-crease in the degree of freedomof atomic vibrations. For themodulationof Rph

th , mainly the following factors compete with each other: (i) surfacescattering effect due to antidots and (ii) interference effect among anti-dots (14). The former effect becomes strongerwith increasing number ofantidots, while the latter effect becomes stronger with increasing lengthof the homogeneous region, that is, the region with a sequence of thesame type of sections (with or without an antidot). Therefore, it is rea-sonable to suppose that the spatial distribution of smaller and largerantidots-distance provides tunability of the balance between the abovetwo effects, although this physical knowledge alone does not explainwhy this particular distribution is optimal; thus, machine learning wasneeded to find the optimal structure.

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The electron transmission functionsQel(E) of representative struc-tures are compared in Fig. 3D. In finite periodic structures, the peaksof Qel(E) corresponding to resonant states appear and split by in-creasing the number of antidots, as shown in fig. S5A (54). Qel(E)of finite periodic structures finally approaches that of periodic antidotstructures, while the residual electrical resistance remains at bound-aries between the periodic antidot region and the semi-infinite pris-tine region, as shown in fig. S5B. Unlike pristine GNRs that have alow Seebeck coefficient due to the absence of the band gap, periodicantidot GNRs have a higher Seebeck coefficient because of the presenceof transport gaps (energy gaps in electron transmission), correspond-ing to energy gaps of an infinite periodic antidot GNR. The introduc-tion of periodic antidots, therefore, can enhance the thermoelectricperformance (Fig. 3A). Thermoelectric performance is further en-hanced by taking advantage of the edge state seen for Qel(E) in−0.078 eV ≤ E ≤ 0.048 eV. Utilization of the edge state requires sup-pression of the electron transmission of resonant states with energynear that of the edge state, as discussed in section S6. In general, anyelectron states are localized, and their transmission is suppressed in in-finite random potential fields (41, 55). Instead, the optimal antidot ar-

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rangement can suppress Qel(E) of target electron states, namely, resonantpeaks in this case. In Fig. 3E, density of states (DOS) and Qel(E) aroundthe energy corresponding to mpeak show that although resonant states ex-ist near the edge state in both the periodic and optimal structures, Qel(E)of the optimal structure is strongly suppressed.

For deeper comprehension of the transmission of the resonantstates, local DOS (LDOS) at the resonant energy is mapped onto eachatom in Fig. 3F, where the top/bottom contours denote the LDOS of thefinite periodic/optimal structure. The indices of resonant state in Fig. 3Fcorrespond to those in Fig. 3E. LDOS in the finite periodic structurespreads over the whole nanostructured region, while the optimal aperi-odic structure leads to the localization of states in limited areas. It is in-tuitively comprehensible that the widely spreading states, which can beregarded as Bloch states, contribute to electron transport, while thestrong localization generates the region with extremely low DOS andsuppresses the electron transport, as shown in Fig. 3E.

Impact of aperiodicityThe above discussions raise the question: How important is the exactarrangement of antidots to control thermoelectric properties? To answer

Fig. 3. Optimization of the antidot GNR structure. (A) Thermoelectric properties—ZT, P, and Rth—of the representative structures: pristine (black), periodic (blue),and optimal (red) structures. The values are normalized by those of the pristine structure. (B) Periodic (top) and optimal aperiodic (bottom) structures. (C andD) Phonon/electron transmission functions, Qph/el(w), of the representative structures. The Fermi level is set to zero for Qel(E). (E) Qel(E) (orange) and DOS in thenanostructured region (green) near the energy levels corresponding to the peak m for ZT, mpeak, of the periodic (top) and optimal (bottom) structures; mpeak =−0.140 and −0.105 eV, respectively, as indicated by dashed lines. The band edge of the edge state is located at E = −0.078 eV corresponding to that of the Van Hovesingularity, and black arrows and their labels indicate resonant states. (F) LDOS distribution of the resonant states of the periodic (top) and optimal (bottom) structures.The resonant numbers correspond to those in (E).

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this question, we calculate the distribution of thermoelectric prop-erties for structures with the same numbers of antidots and homo-geneous regions. Since the number of homogeneous regions is anindication of their lengths, following the above discussion, the twonumbers represent (i) surface scattering effect and (ii) interference ef-fect. The optimal structure has 10 antidots and 7 homogeneous re-gions (Fig. 3B). Figure 4 shows the distributions of P (blue), Rph

th (gray),and Rth (red) of structures with the same numbers as that of the op-timal structure. These three properties are normalized by those of thepristine GNR, and the data are extracted from 359 structures calculatedduring the optimization process. The result shows that, on average,Rth (R

phth ) increases by 3.5 (3.0) times with the optimized arrangement.

In addition, exact replication of the obtained optimal arrangement ofantidots in the experiment may not be required to increase Rth (Rph

th )because of their narrow distribution; the sample SD of Rth (R

phth ) is 0.11

(0.058). Conversely, the wide distribution of Pmakes the arrangementof antidots more important; the SD of P is 0.23, which is 2.1 (3.9)times larger than the SD of Rth (Rph

th ). While P can increase by 3.0times, the improvement could be reduced to 1.3 times without accu-rate arrangement of antidots. The above discussion strongly suggeststhat the control of nanostructures is required for enhancing electrontransport and thus for thermoelectric performance. As for the overalluniqueness of the optima, there are other structures that exhibit ZTinferior to the optimal structure but not by much; as shown in fig. S9,there are 24 structures with ZT > 0.95ZmaxT. These structures havenarrow ranges of numbers of antidots (8 to 12) and homogeneousregions (5 to 7). This result shows that these numbers can be a goodindicator function for high-ZT structures, while we still need the struc-tural optimization because of the fluctuation of thermoelectric proper-ties within structures having the same numbers. It is worth noting thatsuch uniqueness of the optima varies for different models and problems.In cases of model A, there are structures with ZT > 0.95ZmaxT for m =6, 8, and 9, but there are no other structures with ZT > 0.95ZmaxT thanthe optimal structure for m = 7 and 10.

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CONCLUSIONSIn summary, we have shown that the Bayesian optimization techniqueaccelerates structural optimization including multitransport properties,namely,Rth,Rel, and S. The analysis with the periodically nanostructuredGNR reveals the following design strategies for thermoelectric nanostruc-

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tures: (i) The introduction of defects into the entire region of GNR ex-cept for the hexagonal lattices along an edge can improve the powerfactor, and (ii) the labyrinthian shape can increase the thermalresistance. Using the former strategy, we show that introduction of anti-dots can significantly enhance the thermoelectric performance of zigzagGNRs up to 11 (1.2) times larger than that of the pristine zigzag (opti-mal armchair) GNR. Conversely, the careful arrangement of antidots isindispensable for keeping optimized electron states in the nanostruc-tured region; otherwise, the electron properties fluctuate widely.

The state-of-the-art ion beam technique enables the fabrication ofGNRs/antidots with the width/diameter down to 5 nm/1.3 nm (50, 51).While the combination of these techniques may still be challenging atpresent, the fabrication of GNR structures proposed in this studyshould become possible in the near future. Furthermore, we may beable to fabricate arrayed antidot GNR structures combining nanomeshstructures (56) with the above fabrication techniques; because neck re-gions in nanomesh structures can be regarded as arrayed GNRs, theywill become arrayed antidot GNRs if antidots are introduced into theneck regions.

The demonstrated framework enables us to optimize structureswithout any previous knowledge of the relation between the structuresand properties, and with intuitive descriptors to represent the struc-tures. The Bayesian optimization method is expected to be useful forsearches of other geometries and physical properties as long as the num-ber of candidates is processable and the properties are calculable.

METHODSAnalysis methodBefore calculation of thermoelectric properties, a relaxation calcu-lation was performed until the maximum atomic force became lessthan 0.01 meV/Å. To obtain thermoelectric properties (Rth, S, Rel, andthe power factor P = S2/Rel), we calculated the phonon and electrontransmission functions, defined by Qph(w) with the frequency w andQel(E) with the energy E, respectively, using Green’s function methodwith Landauer formalism (27, 42, 44–46, 57, 58) implemented in anAtomistix ToolKit package (see section S1) (59). Here, Rth

−1 =ðRph

th Þ�1 þ ðRelthÞ�1 , where Rph=el

th is the phonon/electron thermalresistance. For electron transport properties, we took the value ata chemical potential (mpeak) within a realistic range, −1.0 eV ≤ m ≤1.0 eV, that gives maximum ZT. For model A, the optimized Tersoffpotential (60) and the p-orbital transfer integral that decreases expo-nentially with increasing C–C bond length (61) were used to calculatephonon and electron transport, respectively. For model B, to performmore realistic calculations, the effects of terminal hydrogen atoms weretreated explicitly. The optimized Tersoff potential was applied to C–Cinteratomic interactions, and the Brenner potential (62) modified fordifferent properties including thermal properties (63) was applied toC–H and H–H interatomic interactions. The transfer integral basedon the density functional theory was used to represent the interactionamong 1s orbital of H and 2s and 2px/y/z orbitals of C (64). All thecalculations were performed under T = 300 K. It is noted that the typicalsystem size in this study (~30 nm) is much smaller than the electronmean free path in pristine graphene or GNRs (~1 mm) at room tempera-ture (65, 66), and thus, electron scattering by phonon was ignored.

Bayesian optimizationBayesian optimization has an advantage over empirical optimiza-tion in its ability to determine required data and a regression curve by

Fig. 4. Fluctuation of thermoelectric properties. P (blue), Rphth (gray), and Rth(red) of structures with the same numbers of antidots and homogeneous regionsas the optimal structure. The thermoelectric properties are normalized by thevalue for the pristine GNR. The fitting curves of Gaussian distributions are shownas a reference.

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machine learning without any previous knowledge of the model. Weused a Bayesian linear regression model for the prediction of ZT

ZT ¼ w⋅ fðxÞ þ e ð1Þ

where x is the vector of the descriptors of GNR structure, f is the featuremap including l basis functions, w is the l-dimensional vector of theweight, and e is the noise subject toGaussian distributionwith themeanof zero and the covariance s2. Bayesian optimization was performed byrepeating the following process: (i) training the Bayesian model with adata set, D ¼ fxi; ZiTgni¼1 , where n is the number of calculatedstructures; (ii) selecting a candidate for the next exact calculation withthe trained model; (iii) performing the exact calculation for theselected candidate; and (iv) adding the result to the data set D for anew training. To obtain f, we adopted the random feature map, whichapproximates the mapping induced by the Gaussian kernel with thewidth h (67) and set l = 5000. The Bayesian linear model with the ran-dom featuremap approximates the Gaussian process, which hasmeritin the flexible expressivity by nonparametric regression. The Gaussianprocess was widely used for materials design with the Bayesian opti-mization (11, 14). The hyperparameters s2 and h were automaticallytuned every 20 optimization runs with the type II likelihood maximi-zation (68). The next point for the exact calculation was determinedfollowing Thompson sampling (69). The above optimization processwas conducted using the open-source library common Bayesian opti-mization (COMBO) (70). More detailed explanations of the Bayesianoptimization method can be found in the studies of Ju et al. (14) andUeno et al. (70).

on June 19, 2020s.sciencem

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SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/6/eaar4192/DC1section S1. Analysis method: Green’s function approachsection S2. Efficiency of single-functional optimizationssection S3. Acceleration of the Bayesian optimizationsection S4. Structural optimization of antidot armchair GNRssection S5. Phonon and electron transport in finite periodic structuressection S6. Effects of resonant states on thermoelectric propertiessection S7. Resonant states in one-dimensional tight-binding chainssection S8. Statistical errors of the efficiency of the Bayesian optimizationsection S9. Uniqueness of the optimal structuresfig. S1. Comparison of optimization efficiency of different functionals.fig. S2. Acceleration of the Bayesian optimization using a topological descriptor: The meanshortest path.fig. S3. Structural optimization of antidot armchair GNR (AGNR).fig. S4. Phonon thermal resistance in finite periodic structures.fig. S5. Electron transport in finite periodic antidot structures.fig. S6. Effects of resonant states on thermoelectric properties.fig. S7. Electron transport properties of one-dimensional tight-binding chains.fig. S8. Efficiency of the Bayesian optimization.fig. S9. Model B structures that exhibit ZT > 0.95ZmaxT.References (71–76)

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Acknowledgments: We would like to thank K. Tsuda, T. Kodama, and T. Shiga for helpfuldiscussions. Funding: This work was financially supported, in part, by the “Materials researchby Information Integration” Initiative (MI2I), Core Research for Evolutional Science andTechnology grant no. JPMJCR16Q5 from the Japan Science and Technology Agency, and theJapan Society for the Promotion of Science (KAKENHI, grant no. 16H04274). Authorcontributions: J.S. directed and designed the project. M.Y. carried out the numericalsimulations. All authors contributed to the discussions and wrote the paper. Competinginterests: The authors declare that they have no competing interests. Data and materialsavailability: All data needed to evaluate the conclusions in the paper are present in thepaper and/or the Supplementary Materials. Additional data related to this paper maybe requested from the authors.

Submitted 7 November 2017Accepted 9 May 2018Published 15 June 201810.1126/sciadv.aar4192

Citation: M. Yamawaki, M. Ohnishi, S. Ju, J. Shiomi, Multifunctional structural design ofgraphene thermoelectrics by Bayesian optimization. Sci. Adv. 4, eaar4192 (2018).

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Multifunctional structural design of graphene thermoelectrics by Bayesian optimizationMasaki Yamawaki, Masato Ohnishi, Shenghong Ju and Junichiro Shiomi

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