スライド 1

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
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
Introduction
Merits
High speed
High integration density
Down scaling of MOSFET
Demerits
(Intel lab)
Various leakage current
High off-state power dissipation
Solutions?
New material,
material New structure,
structure New operation mechanism
2
Our target!
Operation mechanism of conventional MOSFET
Vgate
EF
EC
EV
“OFF” state
Operation mechanism of conventional MOSFET
E
Vgate
EF
EC
f S (E)
EV
“ON” state
I
SS  d (log10 I ) / dVG 
1
Vgate
>60mV/decade at RT
in conventional FET
Degrading by
miniatualizagion
Smaller SS is required!
Armchair and zigzag strain
Stretching in armchair direction
Stretching in zigzag direction
Gap opening by strain
Stratching ratio
armchair
zigzag
z
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
zigzag方向への歪みを加えると・・・
z  0.10 (before gap opening)
t1の値は変わらず
t2のみ小さくなる
z  0.28 (gap opening point)
Eg > 0.0
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
Current density expression
Drain
Source
Calculation model
BZ
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
Pseudo magnetic field due to strain
2
1
B
B
TB approx. and Taylor expanding around the
Dirac point
A
4
6
B
B
A
8
7
→
a２
A
A
B
A
9
B
B
B
A
A
5
A
3
B
A
Energy[eV]
→
a１
TB
Dirac
Band structures from TB and Dirac are
compared
Dirac Hamiltonian is valid for electronic transport
problem
kx/k0
Hamiltonian near the K point
14
Pseudo magnetic field due to strain
Hamiltonian of strained
graphene
strain induced term
→
Hamiltonian of graphene in
magnetic field
：
If A is along x
B = rotA
Shift of kx by strain
Pseudo magnetic field effect
15
Pseudo magnetic field effect on transmission
Unstrained
Graphene
→
Strained Graphene
Electron injection
Evaluating the transmission probability
Along zigzag
strained
Along armchair
歪み有り
y
x
歪み無し
unstrained
歪み有り
strained
y
unstrained
歪み無し
x
16
Pseudo magnetic field effect on transmission
E[eV]
Zigzag strain
Fermi surface
Always transmitted
Armchair strain
ky
Transmission can be blocked
kx
zigzag方向
Eigenstate
armchir方向
17
Transmission analysis based on atomisticTB
Armchair
strained
unstrained
Armchair direction
1.0
Left
t pp r   t0e3.37r / a0 1
Transient region： 4.7 nm
Right
■ πorbital
■ Green’s function
■ atom-atom distance
dependent hopping energy
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0
-0.8 -0.6
ka/
0.0 0.5 1.0
T(E)
-0.8 -0.6
ka/
Band structures and transmission
(ky is fixed at K point)
-1.0
Energy [eV]
Energy [eV]
K point in unstrained K point in strained
ky
kx
Translational symmetry along y direction
 ky is conserved
 Transport gap near E=0
Transmission analysis for
unstrained/armchair strained interface
Unstrained
20% armchair
Strained
Transmission
ky(3a0/2)/pi Transmission
10% armchair strained
25% armchair
Left (unstrained) region
Right (strained) region
Energy
Strained graphene based FET structure at RT
unstrained
strained, and gated
1.0
L=7.4[nm]
armchair strain 10%
S/D
E dope=+0.2eV (p-type)
ID [A/nm]
0.8
0.6
Armchair
0.4
0.2
L=7.4 or 11.7 nm
0.0
-1.0
p-type electrostatic doping
by 0.2eV
10
-0.5
0.0
0.5
1.0
VG [V]
Channel length dependence
Source
Channel
VG
Drain
ID [A/nm]
1
0.1
0.01
L=11.7 nm
1E-3
EF
0.2 eV
1E-4
armchair strain 10%
S/D
E dope=+0.2eV (p-type)
L=7.4 nm
EF  eVDS 1E-5-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0
VG [V]
VDS  0.05 V 
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-1.0
I D   I D k y 
ky
I D k y  
Source
ky

2e
dE T E , k y  f S ( E )  f D ( E )
h 
Channel
ky

L=11.7 nm
15% armchair strained
ID [A/nm]
ky dependent current
-0.5
0.0
VG [V]
Drain
ky
k yc
kx
Current starts to flow from finite ky
equal to the radius of the Fermi circle in
the electrodes.
VG eV
0.5
1.0
k y / k0
Strain dependence and switching performance
VG [V]
1
0.1
ID [A/nm]
ID [A/nm]
1.0
0.9 p-p-p
p-n-p
0.8
0.7
0.6
0.5
0.4
0%
0.3
5%
L=11.7 nm
10%
0.2
armchair strained
15%
0.1
0.0
-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0
10% Armchair strained
L=11.7 nm
42 mV/decade
0.01
L=7.4 nm
1E-3
1E-4
1E-5
0.0
14 mV/decade
0.1
0.2
0.3
0.4
VG [V]
10
ID [A/nm]
1
0.1
0.01
0%
5%
10%
15%
1E-3
1E-4
L=11.7 nm
1E-5 armchair strained
1E-6
-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0
VG [V]
#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
ky
ky
ky
k yc
Channel
Source
EF
kx
Drain
Edoping
EF
Band-to-band tunneling current due to finite ky
f (E  )
Edoping
Shear strained and shear+armchair strained
1.0
p-p-p
p-n-p
ID [A/nm]
0.8
0.6
L=11.7 nm
8% armchair strained
p-p-p
p-n-p
0.8
ID [A/nm]
1.0
Shear strain
0%
5%
10%
15%
0.6
Shear strain
0%
10%
15%
0.4
0.2
L=11.7 nm
0.4
0.2
0.0
-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.0
-1.0
-0.5
VG [V]
0.1
1E-4
Shear strain
0%
10%
15%
L=11.7 nm
1E-5
-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0
VG [V]
1.0
1
0.1
0.01
1E-3
1E-4 Shear strain
0%
1E-5
5%
1E-6 10%
1E-7 15%
1E-8
-1.0
-0.5
ID [A/nm]
ID [A/nm]
1
1E-3
0.5
VG [V]
10
0.01
0.0
L=11.7 nm
8% armchair strained
0.0
VG [V]
0.5
1.0
Effect of atomic position fluctuation
unstrained
Armchair
Total transmission
12
Super cell containing
54 UCs along ydirection
8
6
4
2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.0
ky(3a0/2)/
Armchair strained
(with atomic position
fluctuation)
10
0
-1.0
L=12 nm
10% along armchair
+0.2eV doping in S/D (p-type)
-0.5
0.0
0.5
No fluctuation
0.1 Ang z
0.1 Ang y
1.00.1 Ang x
E-EF [eV]
Band st. in S/D
-0.5
0.0
E-E [eV]
0.5
1.0
Total transmission
Total transmission
10
10% along armchair
+0.2eV doping in S/D (p-type)
12
10
fluctuation along
z,y,z, with amplitude
0.1Angstrom
8
6
4
2
0
-1.0
-0.5
0.0
0.5
10% along armchair
+0.2eV doping in S/D (p-type)
8
fluctuation
along
x with various
amplitudes
6
4
No fluctuation2
0.1 Ang z
0.1 Ang y
0
1.00.1 Ang x -1.0
-0.5
E-EF [eV]
Total transmission
Total transmission
0.01
1E-4
No fluctuation
0.1 Ang z
0.1 Ang y
0.1 Ang x
1E-6
1E-8
1E-10
1E-12
-1.0
-0.5
0.0
E-EF [eV]
0.5
1.0
E-EF [eV]
10% along armchair
+0.2eV doping in S/D (p-type)
1
0.0
No fluctuation
0.1 Ang x
z
x
0.2 Ang z
0.3 Ang x
z
0.4 Ang x
z
0.5
1.0
10% along armchair
+0.2eV doping in S/D (p-type)
100
1
0.01
No fluctuation
0.05 Ang z
0.1 Ang z
0.2 Ang z
0.3 Ang z
0.4 Ang z
1E-4
1E-6
1E-8
1E-10
1E-12
-1.0
-0.5
0.0
E-EF [eV]
Strain induced atomic position shift  0.07 ang for 10% strain
0.5
1.0
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 subthreshold swing with SS< 60mV/decade due to the bandto-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
全エネルギーのSR依存性
破断限界
バンドギャップのSR依存性
Transmission analysis for
unstrained/shear strained interface
Unstrained
20% shear
Strained
ky(3a0/2)/pi Transmission
Transmission
10% shear strained
25% shear
Left (unstrained) region
Right (strained) region
Energy
Strain along armchair direction
・armchair方向に歪みを印加したグラフェンに電子を入射
1
歪み有り
E = 0.1[eV]
E = 0.5[eV]
E = 1.0[eV]
0.8
歪み無し
0.6
y
透過率Ｔ
x
透過率
0.4
0.2
(
)
0
0
0.02
0.04
0.06
0.08
0.1
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
基本並進ベクトルの変化
zigzag方向への歪み
単軸歪み
armchair方向への歪み
せん断歪み (shear)
shear + zigzag方向への歪み
shear + armchair方向への歪み
1つのパラメータで制御するモデル
2軸性歪み
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
遷移領域
歪み無し
せん断歪み
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
1.0
4
3
2
1
0
kx
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1
面直揺らぎ
5
z=0
0.3 Ang
0.5 Ang
1.0 Ang
A
1.0
0.5
Total transmission
Total transmission
5
Energy [eV]
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）
ky
Right
4
3
2
0
ka/
1 0 2 4 6
-1
T(E)
面内揺らぎ
xy=0
0
ka/
1
Energy [eV]
Armchair
方向
Left
-1.0
0.1 Ang
0.3 Ang
0.5 Ang
1
0
-0.4
-0.2
0.0
0.2
0.4
-0.4
-0.2
0.0
0.2
0.4
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
Energy [eV]
Energy [eV]
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
A
Pseudo magnetic field due to strain
armchair方向に歪み
a rm ch a ir
zigzag方向に歪み
y
x
z ig z a g
y
・強束縛近似法
・K点付近で近似
x
歪み無し
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
歪み有り
歪み無し→歪み有りへの
電子の透過率を求める
36
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
y
歪みを印加したハミルトニアン
x
歪み無し
歪み有り
エネルギー固有値、固有ベクトル
1.2
確率流密度
1
0.8
透過率Ｔ
透過率Ｔ 0.6
0.4
透過率
0.2
T
{
0
0
0.02
0.04
0.06
0.08
0.1
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
Ay  iAx    t r, n e
iK n
n
A
A

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