RPGR2013 10p-A3-O3 Sep.10, 2013 Tokyo Institute of Technology Effect of lateral strain on electronic transport in graphene: interplay between band gap formation and pseudo magnetic field effect Satofumi Souma, Masayuki Ueyama, Matsuto Ogawa Department of Electrical and Electronic Eng. Kobe University, Japan Related presentation: 12a-P3-16 T.Funatani, T.Yamamoto, K. Sasaoka, M. Ogawa, S. Souma Effect of in-plane strain on thermal properties of graphene

Use of external strain to modulate the electronic properties Background Absence of bandgap at the Fermi energy in graphene  drawback for the FET application Zero bandgap Use of external strain to modulate the electronic properties Effect of lateral strain in single layer graphene 1. Band gap opening at the Dirac point --- Large strain more than ~20%, Conventional Semiconductor like behavior (G.Cocco, E.Cadelano, and L.Colombo, PRB81, 241412R (2010)) 2. Pseudo magnetic field due to local strain (V.M.Pereira and A.H.Castro Neto, PRL 103, 046801 (2009)) ＊Perturbative effect based on the Dirac particle picture, induced even by small strain. *No bandgap opening. *QHE in zero magnetic field, *Valley filtering effect. * How these two effects are connected? * How realistic is the pseudo magnetic field effect for device application?

Requirement from application view point

Operation mechanism of conventional MOSFET “OFF” state

Operation mechanism of conventional MOSFET “ON” state >60mV/decade at RT in conventional FET Smaller SS is required! Degrading by miniatualizagion

Armchair and zigzag strain Stretching in armchair direction Stretching in zigzag direction

Gap opening by strain Stratching ratio armchair zigzag z Stratching ratio Zigzag strained case Gap opening for the strain>0.28, and Eg increases with strain Armchair strained No bandgap opening

Interpretation of bandgap opening t1=t2=t3 for unstrained case zigzag direction Eg > 0.0 zigzag方向への歪みを加えると・・・ t1の値は変わらずt2のみ小さくなる z = 0.10　(before gap opening) z = 0.28　(gap opening point) Gap opening if t2<t1/2

Efficient gap opening by shear strain shear strain in graphene Gap opening for z>0.13 Ratios of the hopping energies are z = 0.10　(before gap opening) z = 0.13　(gap opening point) Hopping energies are efficiently modulated

Calculation of FET performance at 0K Calculation model Current density expression BZ Source Drain L decreases  tunneling effect  current density at VG (eff) ＝0.0 increases T = 0.0K, VDS = 0.01V, L = 46.8nm（ zero strain）

Calculation of FET performance at 0K armchair strain zigzag strain

Calculation of FET performance at 0K shear strain shear + armchair strain Still large strain is required

Dirac Hamiltonian is valid for electronic transport problem Pseudo magnetic field due to strain TB approx. and Taylor expanding around the Dirac point Band structures from TB and Dirac are compared TB Dirac Energy[eV] Dirac Hamiltonian is valid for electronic transport problem kx/k0 Hamiltonian near the K point   14

→ Pseudo magnetic field due to strain Pseudo magnetic field effect   Hamiltonian of strained graphene   Hamiltonian of graphene in magnetic field B = rotA ： If A is along x strain induced term   → Shift of kx by strain Pseudo magnetic field effect 15

→ Pseudo magnetic field effect on transmission Unstrained Graphene Electron injection → Evaluating the transmission probability Along zigzag Along armchair strained unstrained unstrained strained 16

Transmission can be blocked Pseudo magnetic field effect on transmission Always transmitted Transmission can be blocked Zigzag strain Armchair strain Fermi surface zigzag方向 armchir方向 Eigenstate 17

Transmission analysis based on atomisticTB ■　πorbital ■　Green’s function ■　atom-atom distance dependent hopping energy Armchair strained unstrained Armchair direction Transient region： 4.7 nm K point in unstrained K point in strained Translational symmetry along y direction  ky is conserved  Transport gap near E=0 Band structures and transmission (ky is fixed at K point)

Transmission analysis for unstrained/armchair strained interface Left (unstrained) region Right (strained) region ky(3a0/2)/pi Energy

Strained graphene based FET structure at RT unstrained strained, and gated Armchair L=7.4 or 11.7 nm p-type electrostatic doping by 0.2eV Channel length dependence Source Channel Drain

ky dependent current Source Channel Drain Current starts to flow from finite ky equal to the radius of the Fermi circle in the electrodes.

Strain dependence and switching performance #Very steep SS=14mV/decade at RT for the strain > 8% #On/Off ration can be as high as five orders of magnitude

Mechanism of steep SS in p-n-p regime Drain Channel Source Band-to-band tunneling current due to finite ky

Shear strained and shear+armchair strained

Effect of atomic position fluctuation unstrained Armchair L=12 nm Armchair strained (with atomic position fluctuation) Band st. in S/D Super cell containing 54 UCs along y-direction

Strain induced atomic position shift  0.07 ang for 10% strain fluctuation along z,y,z, with amplitude 0.1Angstrom fluctuation along x with various amplitudes x Strain induced atomic position shift  0.07 ang for 10% strain

Summary #1 Armchair-strain induced pseudo magnetic field effect can be applied to realize graphene based FET. #2 PNP (or NPN) mode allows us to obtain steep sub-threshold swing with SS< 60mV/decade due to the band-to-band-tunneling transistor like operation (without changing the bandstructures in each region). #3 Shear-type strain can help to enhance the effect of pseudo magnetic field effect. #4 Pseudo magnetic field induced FET operation is robust against the atomic position fluctuation unless the fluctuation amplitude is comparable to the external strain.

Stretched AGNR 破断限界 全エネルギーのSR依存性 バンドギャップのSR依存性

Transmission analysis for unstrained/shear strained interface Left (unstrained) region Right (strained) region ky(3a0/2)/pi Energy

Strain along armchair direction 透過率Ｔ   透過率 ( ) 歪み率γ 歪み率　大 エネルギー　大 透過率　減少 透過率の減少　緩やか 31

面内歪みによるギャップ誘起 単軸歪み 2軸性歪み 1つのパラメータで制御するモデル 基本並進ベクトルの変化 zigzag方向への歪み armchair方向への歪み せん断歪み (shear) shear + zigzag方向への歪み 2軸性歪み shear + armchair方向への歪み 1つのパラメータで制御するモデル

Calculation of FET performance at 0K ・歪み印加による電流の特異な変化。 ・ギャップ誘起の前段階の領域でも電流による歪の検知が可能

せん断歪みによるzigzag方向の伝導制御 歪み無し 遷移領域 Zigzag 方向 左右領域でのバンド構造 （全てのkyについてplot） 透過率 Energy 右領域の Eg y方向の波数 ジグザグ方向への輸送についても、擬似磁場効果を援用した電子透過制御が可能

Effect of atomic position fluctuation Armchair 方向 この領域の原子位置を ランダムに微小変化 （幅方向に５４UCから なるスーパーセルで計算） 面直揺らぎ 面内揺らぎ 面直揺らぎ→ランダムなベクトルポテンシャルの方向がArmchair歪みによるベクトルポテンシャルの方向に一致 Armchair方向の輸送への影響が小さい

Pseudo magnetic field due to strain zigzag方向に歪み armchair方向に歪み ・強束縛近似法 ・K点付近で近似 局所的に歪ませたグラフェンの ハミルトニアンを求める 歪み無し→歪み有りへの 電子の透過率を求める 36

・zigzag方向に歪みを印加したグラフェン Strain along zigzag direction ・zigzag方向に歪みを印加したグラフェン 　に電子を入射 エネルギー固有値、固有ベクトル 歪みを印加したハミルトニアン 確率流密度 透過率Ｔ 透過率Ｔ 透過率   T { 歪み率γ 歪み率に関わらず全透過 37

まとめ ・せん断歪みを印加する事により効率的なギャップ誘起 ・歪み印加による特異な電流変化 ・局所歪みによる擬似磁場に起因する電子透過制御 　　→原子位置の揺らぎによって劣化するが、 　　　面直揺らぎの影響は少ない ・ジグザグ方向への輸送に関しても、せん断歪みを用いる事で 　擬似磁場の効果を援用した電子透過制御が可能。 　バンドギャップ効果と擬似磁場効果の共存。

Strain induced modulation of Bandstructure for various transverse wavenumber ky zigzag strain shear strain shear + armchair strain

面内歪みによるギャップ誘起 ある距離までの原子軌道同士の重ね合わせは考慮し、それ以上の距離では無視するバンド計算法 グラフェンのバンド構造 この時のハミルトニアンは

局所歪みによる擬似磁場効果 グラフェンに歪みを印加したとき、最も効率良くギャップを開かせる変形は、　　　　　　　　　　　　　せん断歪みとarmchair方向への歪みを組み合わせたものである 3方向のホッピングエネルギーのうち、2つを同じ比率で小さくし、残りの1つを大きくすれば、効率良くギャップを開かせることが出来る グラフェンをＦＥＴのチャネル材料として用いたときの各歪みに対する電流・電圧特性計算を行った zigzag方向,せん断歪み,せん断+armchair方向への歪みを印加した場合、ギャップの大きさに対応した電流密度が0となる領域を得ることが出来る　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 →　スイッチング特性（明確なOFF領域が得られる） 各歪み印加時は、歪み率の増加に伴い電流密度が変化（増加, 減少）する　　　　　　　　　　　　　　　　　　 →　圧力センサへの応用 今後の課題 ・ポアソン比（縮み）の導入 ・ギャップと歪みの関係に対する更なる理解 ・有限温度での電流・電圧特性の計算