Chapter 2 Epoch-Making Simulation

Large Scale Simulations for Carbon Nanotubes
Project Representative

Syogo Tejima

Research Organization for Information Science & TechnologyAuthors

Syogo Tejima* , Satoshi Nakamura* , Yoshiyuki Miyamoto* , Yoshikazu Fujisawa* and1Hisashi Nakamura*1123*1 Research Organization for Information Science & Technology (RIST)*2 R&D Unit Central Res. Labs. Green Innovation Research Laboratories, NEC Corporation*3 Honda R&D Co., Ltd.
CARBON NANOTUBE RESEARCH GROUP

Morinobu Endo, Eiji Osawa, Atushi Oshiyama, Yasumasa Kanada, Susumu Saito,Riichiro Saito, Hisanori Shinohara, David Tomanek, Tsuneo Hirano, Shigeo Maruyama,Kazuyuki Watanabe, Takahisa Ohno, Yutaka Maniwa, Yoshikazu Fujisawa,
Yoshiyuki Miyamoto, Hisashi Nakamura
Nano carbon materials as nanotube, fullerene and graphen have a potential for applications to the advanced industries. For nanocarbon materials, it has been recognized as large-scale simulation is a powerful and efficient tool to find and create new functional
nano carbon materials.
Aiming at conducting the productive simulation for nano-materials, we have developed the large-scale simulation models such astight-binding molecular dynamic model, ab-initio density functional theory (DFT), and time-dependent DFT model.
In this term, by utilizing these models effectively, we have studied various physical properties of nano-carbon and applications
such as (1) application of novel functions of Mackay Crystal to solar cell, (2) Large-scale Simulation on Electron Conduction inCarbon Nanotubes at Finite Temperature, (3) applications of time-dependent density functional theory for photo-chemical reactionof molecules inside CNTs. Along these works, we have realized as the Earth Simulator is a very powerful tool for large-scale nanomaterial simulations.
Keywords: Large scale simulation, TB theory, ab initio theory, Time-dependent DFT, Carbon Nanotube, Fullerenes, Graphene,
Mackay crystal, solar cell, Green energy, quantum electronic transport, photoelectric material

1. INTRODUCTION

which there are three primary objectives as (1) design of

Nano-carbon materials have been expected to bringinnovative nonmaterial with the required properties; (2) obtainbreakthrough to material science and nanotechnology. A lot of

fundamental properties in nano-scale matter, and (3) develop

potential applications of nanotube and fullerene to electronic

new applications.

devices have been attracted to scientists and engineers.

2. Physical studies on nano materials
2.1 Application of Novel Functions of Mackay Crystal to
Solar Cell [1]In the present days, large-scale numerical simulation byusing supercomputer's computational performance has turned to
be a very efficient tool and leverage for investigating their novel
material properties. It now allows us to simulate complex nano-We have carried out the extensive simulation on the Mackaystructures with more than ten thousand atom of carbon

crystals using GSW method to find synthesis process and DFT

Aiming at using large-scale simulations on the Earthapproximation to obtain mechanical and electrical properties. In

Simulator, we have developed an application package of2009, from the investigation on electronic properties of Mackay

ab initio DFT theory and parameterized tight-binding (TB)

crystals, it was indicated that the band gap of Mackay crystals ofmodels. Especially, the TB model shows that it is very suitable

P48, P144 and P192 are ranged from 0.05eV to 0.94 eV. On thefor the very large systems even if it has a lack of symmetricalbasis of this prediction, we proposed that a tandem-type solar

arrangement.

cell could be conceptually designed by stacking Mackey crystal

In this term, we have carried out simulation studies, infilms with the different size. This solar cell would be able to125

Annual Report of the Earth Simulator Center April 2010 - March 2011

Fig. 1 A carbon surrounded only hexagon is replaced by Boron andNitrogen atom. Replaced energy is described at the right side.

Fig. 2 A carbon surrounded two hexagons and a octagon
are replaced by Boron and Nitrogen atoms. Replacedenergy is described at the right side.

Fig. 3 The N-doping leads to injection ofelectron in the energy band.

Fig. 4 The B-doping leads to injection of holein the energy band.

absorb the sun light with near infrared light.

efficiency. Mackay crystals are expected to be a highly efficient

The electrical energy is converted from photon energy ofphotoelectric material for solar cell.

solar cell. In order to derive electrical energy from the Mackay
crystal, we have to set p-n semi-conductance junction. In thisyear, we have simulated the electronic structure of Mackay

2.2 Large-scale simulation on electron conduction of carbon-nanotubes in finite-temperature environments [2]crystals with Nitrogen and Boron as the doping material for

The rapid development of integrated circuit technology isproducing p-n Mackay junction. Nitrogen and Boron inject

primarily achieved due to the sustained downscaling in theelectrons and holes into the crystal as the carrier mobility,

metal-oxide-semiconductor field effect transistor (MOSFET).

respectively, into the crystal. Figures 1 and 2 show the energy-

Moore's law is a phenomenological observation that the number

dependence of doping materials inserted to different atomicof transistors on integrated circuits doubles every two years.

sites. From these figures, it is clear that the doped atomic site is

Silicon-based MOS technology, however, is considered to facesurrounded by only hexagons and by two hexagon and octagon

technical and economical limits as soon as at the end of thisin fig. 3, respectively. The replaced energy measured from puredecade. Therefore, new types of nanoscale devices are being

crystal is also exhibited at the right side in Figs. 1 and 2. The

investigated aggressively. In this context, a lot of attention issimulation has revealed the doping of Nitrogen and Boron are

paid to integrated circuits made of carbon-nanotubes (CNTs).

more stable at octagonal than hexagonal one.

In nano-scale systems, the macroscopic Ohm's law breaks down

Figures 3 and 4 show the electrical density of state fordue to the various effects caused by finite size effects. Here one

Nitrogen and Boron doping to the octagonal site. The N-doping

needs to study the transport behavior of CNTs using a quantum(B-doping) leads to the injection of holes (electrons) as shown inmechanical description.

Fig. 3 (Fig. 5). Such as band structure indicates N- and B-doping

Our group has been developing an order-N code forto Mackay crystal can lead to a carrier of hole and electrons insimulating electron conduction. Our approach is based on thethe crystal.

non-equilibrium Green's function (NEGF) formalism which

As the next step, from valence-conduction band transitionis the method of choice to evaluate electron conductance inprobability, we plan to simulate the photon energy absorption

nanostructure.
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Chapter 2 Epoch-Making Simulation

J j → j = 2e12h×

∑ t j ξ j ξ ({R})∫{ f (ɛ– μ R) – f (ɛ– μ L )}possible. As an application of our substitute charge method, wḙ

shown in Fig.5 (a), where the gate voltage is 2.0 eV and the bias

2 21 1consider the system consisting of one gate and two electrodes asξ 1ξ 2

̭a∑ Re [ĜrCC (ɛ){–iГ (ɛ)}G CC(ɛ)]( j ,ξ ,σ ) ( j ,ξ ,σ) dɛLLσ1

122

applied between two electrodes is 1.0 eV. Then, the effect of the,gate and electrodes is represented by the substitute charges asshown in Fig. 5 (b).

where the retarded and advanced NEGFs are defined as follow:r ( ɛ) = =ĜCCÎ CC̭̭̭rr( ɛ + iγ )I CC – H CC – ΣLL( ɛ) – ΣRR (ɛ)̭  a ( ɛ) = =ĜCCNow, in order to simulate microscopic electron conduction
at atomic scale using our method, we need to calculate NEGFs  Î CC̭̭̭aa( ɛ + iγ )I CC – H CC – ΣLL( ɛ) – ΣRR (ɛ)

of which number is equal to the number of mesh points used,in numerical integration with respect to energy. On the one
hand, the accuracy of calculation increases with the increasḙ    of the number of mesh points; on the other hand, the number.

of mesh points needed to obtain reliable estimates of theelectron conduction increases inversely proportionally withThe true electron conduction is obtained by taking the limit

the strength of nanotube-electrode junction. As a result, whenas g tends to positive zero. In order to calculate the NEGFs

the strength of the junction is comparatively small, it takes

in the above equations, we have adopted the embedding

enormous computing time to simulate the electron conduction.

potential method which works well on treating quasi-one-

To overcome this difficulty, we have developed a method for

dimensional systems such as carbon-nanotubes. This method

computing the NEGFs with high accuracy and high speed byprovides an algorithm of the order-N, indicating the possibility

applying analytic continuation via the Pade approximants.

of performing high-performance computation in large-scale

An example of our method is shown in Fig. 6. We first havemodels. The NEGF formalism can also be applied to CNTs of

simulated NEGFs at g = 0.00094278 eV through the embedding

arbitrary chirality or CNTs with defects.

potential method and computed the electron current indicated byMoreover, by solving self-consistently the coupled NEGF

the green circle. Using these NEGFs as input data of the Padeand three-dimensional Poisson equations, we can obtain theapproximants, we obtained the electron currents represented

electrostatic potential distribution in carbon-nanotubes. In thisby the blue triangles. Finally, applying the Pade approximants

case, we need to solve the Poisson equation as fast as possible.

again to the values of the electron current corresponding toA promising method for this is to use the substitute charge

different values of g, we obtained the ideal results which aremethod. However, since the conventional substitute charge

obtained in the limit as g tends to positive zero (the red lozenge

method determines the values of the charges on the basis ofin Fig. 6).

empirically-predicted charge distribution, this method has

According to the simulation results using 80 nodes (640the problem that its calculation accuracy depends on how toprocessors) on the Earth Simulator, it took about 1,800 sec toselect the substitute charge distribution. Therefore, we havecompute all the NEGFs with out method. On the other hand,developed a new method by changing not only the values ofwhen all the NEGF is computed directly without the Padethe charges but also their positions. Our method allows us toapproximants, it took about 5,100 sec to perform the same

impose the Dirichlet boundary conditions on three-dimensional

simulations. Hence, our method is about 2.8 more effective thansurfaces with arbitrary shape and to treat even problems withthat without the Pade approximants.

incomplete boundary conditions. In addition, once the substitutecharge distribution is determined, there is no need to repeat thisprocedure any more. This makes efficient and stable calculation

(a)

(b)

Fig. 5 (a) Schematic view of the system consisting one gate and two electrodes; (b) the different
color dots represent different values of the substitute charge.

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Annual Report of the Earth Simulator Center April 2010 - March 2011

this molecule can be disintegrated as following figure with ultrashort femto-second laser pulse with corresponding power is
1.91x1015W/cm2.
This result will stimulate experimental work in nanotube
community and been published in Physical Review Letters in
2010, as well as the last year's product (laser-induced grapheneexfoliation). Because of the mass difference, only H atom
obtained high kinetic energy which is enough to move outsidefrom the open-end of the CNT into the vacuum. We expect that
pulse laser can extract from any molecule that contains H atom
being encapsulated by CNT.
Fig. 6 The green circle was calculated via the embedding potential
method. The blue triangles were evaluated from the Padeapproximants of which input data are the NEGFs at g =0.00094278 eV. The red lozenge is estimated using the value ofthe blue triangles as input data of the Pade approximants.

This year, we are exploring more complex situation, i.e., amolecular dimer of acetylene which never exists in natural gas
phase but can exist in CNT. Pulse-induced synchronized motionof two molecules inside CNT is monitored by the simulation.
We expect that further time-simulation will reveal a new path of

2.3 Application of time-dependent density functional
theory for photo-chemical reaction of molecules
inside CNTs [3]chemical reaction of these molecules.

3. SUMMARY

Following 2009, application of femto-second laser was

Large-scale simulations have been carried out on

investigated for photo-chemical reaction of molecules inside

nonmaterial by using ab initio density functional theory andCNTs. Since molecules in gas phase are dilute and thus has

the parameterized tight-binding models. These optimized

low cross-section of photo-irradiation, confining molecules intomodels allowed us to simulate the nano maretial properties with

nano-space was expected effective to increase the cross-section.

excellent performance on the Earth Simulator. It enables us toA question here is whether there is proper material which cancome across discoveries of novel phenomena in nano scale andconfine molecules in narrow space and can penetrate opticalto find out some useful materials for clean energies and nanofield. We tested semiconducting (8,0) CNT, and discovered

device.

this CNT works as nano-test-tube. We performed electron-ion
dynamics under present of pulse laser with alternating electric

References

field mimicking the laser pulse by solving time-dependent[1] S. Tejima, Nikkei newspaper, Sep 30 (2010).

Kohn-Sham equation[2] S. Nakamura,et al., NT10 Eleventh International
Conference on the Science and Application of Nanotubes,

iħdΨn (r, t)= {HKS (r, t) + Vext (r, t)}Ψn (r, t) ,dt

Montreal, June 26 (2010).
[3] S. Miyamoto, et al., Phys Rev lett Vol.105 pp248301-1248301(2010).

which includes scalar potential mimicking optical field.
An HCl molecule was chosen as our case study, which is
historically studied as a subject of photo-induced disintegration.
When this is encapsulated inside (8,0) nanotube exothermically,

Fig. 7 Dynamics of an HCl inside (8,0) CNT.

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Chapter 2 Epoch-Making Simulation

カーボンナノチューブの特性に関する大規模シミュレーション
プロジェクト責任者

手島正吾

高度情報科学技術研究機構

著者

手島正吾 * ，中村1

賢 * ，宮本1

良之 * ，藤沢2

義和 * ，中村3

壽*1*1 高度情報科学技術研究機構*2 NEC R&D ユニット 中央研究所 グリーンイノベーション研究所*3 本田技術研究所
1. 研究目的
優れた物性が予想されナノテクの基本材であるナノ炭素類（カーボンナノチューブ（CNT）、フラーレン、グラフェン）の電子・機械特性を従来不可能だった大規模シミュレーションにより推定し、科学技術及び産業界に提供すると共に、応用として特性の優れた新構造を発見し、基本材の拡充に寄与する目的で実施された。
2. 成果
これまで、我が国のナノチューブ研究で当面する課題解決に向けた応用シミュレーションを実施した。本年度さらに、グリーンエネルギーに関するシミュレーションを推進した。得られた結果を以下に示す。
（1）これまでマッカイ結晶の包括的な特性把握のために、合成法、機械特性などのシミュレーションを実施してきた。
昨年度は、電子構造特性のシミュレーションにより、マッカイ結晶が太陽電池材料としての可能性があることが示せた。本年度は、光電変換で生じた電気エネルギーを外部に取り出す為に、キャリアを持つを p-n 半導体を作りその特性を調べた。ホールと電子のキャリアを挿入するためのドーピング元素としてホウ素と窒素を選んだ。全エネ
ルギー計算の結果、ドーピング元素がマッカイ結晶の８員環の原子の一部を置換し易いことが明らかになった。また、置換された結晶の電子状態密度から、ホウ素置換では価電子帯にホールが、窒素置換では伝導体に電子がそれぞれ出現することが明らかになった。キャリアが導入されことにより、太陽電池として n-p 型半導体ジャンクションを利用して電流を外部に導く機能を得る事ができた。積層半導体マッカイ結晶は、応用として、広範囲の波長の光吸収を可能とする、高効率太陽が期待される。
（2）本研究では、非平衡グリーン関数（NEGF）法を用いた伝導特性計算において、CNT のような擬 1 次元系で威力を発揮する埋め込みポテンシャル法を用いたアルゴリズムを採用し、オーダー N 法の量子伝導シミュレーションコードを開発している。NEGF 法では、ポテンシャル分布を求める際にポアソン方程式を高速に解く必要がある。そこで、電荷の値だけでなく電荷の位置も変化させることができる新たな電荷代用法を開発し、これにより高速で安定した計算を可能にした。これらの手法を用いて原子スケールでの微視的な電流をシミュレーションするには、エネルギーに関する数値積分での分点の数だけ NEGF を計算する必要がある。高速化を目的に、パデ近似による解析接続を用いて NEGF を高い精度で高速に求める計算手法を開発し、計算時間を 3 分の 1 程度に減らすことに成功した。
（3）ナノチューブ内分子分解反応の第一原理時間発展計算によるシミュレーション実施した。フェムト秒レーザショッ
トの強さを制御するとナノチューブ内部で HCl 分子は自然に分解する現象が生じた。この結果は、カーボンナノチューブを使って分子の光化学反応の過程が進行する事を示している。
キーワード : 大規模シミュレーション , タイトバインディング理論 , 時間依存密度汎関数法 , カーボンナノチューブ ,グラフェン , マッカイ結晶 , 太陽電池 , グリーンエネルギー , 量子伝導 , 光電材料 , 光化学反応

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