大規模第一原理電子状態計cpu6 cpu7 cpu8 cpu3 cpu4 cpu5 cpu1 cpu2 natori’s compact model...
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![Page 1: 大規模第一原理電子状態計CPU6 CPU7 CPU8 CPU3 CPU4 CPU5 CPU1 CPU2 NATORI’S COMPACT MODEL FOR BALLISTIC Si NANOWIRE MOSFET K. Natori, IEEE Trans. Electron Devices 55, 2877](https://reader036.vdocuments.site/reader036/viewer/2022081403/60ae2b9ca6715062146f1e19/html5/thumbnails/1.jpg)
東京大学大学院工学系研究科 岩田潤一 �
大規模第一原理電子状態計算手法の開発とシリコンおよび炭素ナノ物質への応用 �
課題番号:hp120245 �
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¡ 量子力学的第一原理シミュレーション § 密度汎関数法 § RSDFT(実空間差分法)
¡ Siナノワイヤ(次世代トランジスタの材料)
¡ 櫻井 -杉浦法のバンド構造計算への応用(筑波大 二村保徳)
¡ 捩じれた二層グラフェン(東大 内田和之) ¡ RSDFT-CPMDの実装(東大 小泉健一、阪大 重田育照)
内容 �
CPMD = Car-Parrinello Molecular Dynamics �
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RSDFT �
FIRST-PRINCIPLES DFT CALCULATION IN REAL-
SPACE GRID METHOD �
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DENSITY FUNCTIONAL THEORY �
E[{!i}]= dr!
i=1
N
" !i
*(r) #
1
2$2!
i(r)
%
&'
(
)*
+ dr! !(r)vion(r)
+1
2dr! d +r!
!(r)!( +r )
r# +r
+EXC[!]
!(r) = "i(r)
2
i=1
N
! P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864.
W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133.
ion�
unit cell�
electron �
Energy functional�
Electron density�
電子−イオン相互作用�
運動エネルギー�
電子間クーロン相互作用�
その他量子力学的効果 (交換相関効果)�
自然界で実際に安定に存在する状態 → エネルギー最小の状態
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KOHN-SHAM EQUATION �
!E["]
!"i
*(r)
= 0 dr! !i*(r)!i (r) =1
)()()(21 2 rrr iiiKSv φεφ =⎟
⎠
⎞⎜⎝
⎛ +∇−
∑=
=N
ii
1
2)()( rr φρ vKS(r) = d !r!
!( !r )
r! "r+!E
XC
!"(r)+ v
ion(r)
汎関数最小化�
constraint�
系の最安定状態(基底状態)�
変分方程式:Kohn-Sham方程式�
電子密度�
ポテンシャル�
W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133.
非線形固有値問題�
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REAL-SPACE FINITE-DIFFERENCE PSEUDOPOTENTIAL METHOD �
)()()(][)(ˆ
21 2 rr
rrrrr jjj
XCion
Edv φεφδρ
ρδρ=⎟
⎟⎠
⎞⎜⎜⎝
⎛+
ʹ′−
ʹ′ʹ′++∇− ∫
Kohn-Sham equation is solved in discretized space �J. R. Chelikowsky et al., Phys. Rev. B50, 11355 (1994). J.-I. Iwata et al., J. Comp. Phys. 229, 2339 (2010).
FFT free �
u MPI ( Message-Passing Interface ) library MPI_ISEND, MPI_IRECV → finite-difference calc.
MPI_ALLREDUCE → global summation u OpenMP
Further grid parallelization (within each CPU) is performed by thread parallelization
CPU0
CPU8 CPU7 CPU6
CPU5 CPU4 CPU3
CPU2 CPU1
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NATORI’S COMPACT MODEL FOR BALLISTIC Si NANOWIRE MOSFET �
K. Natori, IEEE Trans. Electron Devices 55, 2877 (2008) �
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BAND STRUCTURE OF SiNW �
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0 0.3061
eigenvalue
k
d!n (k)dk
= "nk ! i" "nk
!12"2 + vSCF (r)+ ik #"+
k2
2$
%&
'
()!nk (r) = !nk!nk (r)
n番目の状態にある電子の速度�
物質のバンド構造の計算:SCFポテンシャルを得た後、波数 k というパラメータを導入し、 パラメータ振りながら固有値問題を解き直す�
(電流に関係する量)�
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CALCULATION PROCEDURE �
!!2
2m"2
+ v(r)#
$%
&
'(!nk
(r) = En(k)!
nk(r)① Perform band calculation(第一原理計算)�
② Give Vg-Vth and Vds as input parameters, and solve the following equation�
Ne
Fore= 2
n
!dk
2!dEn (k )
dk>0
"1
1+ expEn(k)#µ
s
kT
$
%&
'
()
Ne
Back= 2
n
!dk
2!dEn (k )
dk<0
"1
1+ expEn(k)#µ
d
kT
$
%&
'
()
µd = µs ! qVds
Cg Vg !Vth !µs!ECBM
q
!
"#
$
%&=Qe
Channel= q Ne
Fore+ Ne
Back( )
µsGet �
Id = 2qn! dk
2!dEn (k )dk >0" 1
!dEn (k)dk
1
1+ exp En (k)#µs
kT$
%&
'
()+ 2q
n! dk
2!dEn (k )dk <0" 1
!dEn (k)dk
1
1+ exp En (k)#µd
kT$
%&
'
()
③ Calculate the drain current by the following formula�
ソース ドレイン ゲート 電極
SiNW �
(電極の効果はモデルで評価)�
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1.8
1.6
1.4
1.2
1.0
(eV)
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
(eV)
0.300.200.100.00-1.0
-0.8
-0.6
-0.4
-0.2
0.0
(eV)
0.40.30.20.10.0
1.6
1.4
1.2
1.0
0.8
(eV)
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
(eV)
0.150.100.050.00
SiNW(111) diameter:4nm temperature:300K
60
40
20
0
Id (µ
A)
1.00.80.60.40.20.0Vd (V)
VG-VT=0.1V
VG-VT=0.4V
VG-VT=0.7V
VG-VT=1.0V
60
40
20
0
Id (µ
A)
1.00.80.60.40.20.0Vd (V)
VG-VT=0.1V
VG-VT=0.4V
VG-VT=0.7V
VG-VT=1.0V
60
40
20
0
Id (µ
A)
1.00.80.60.40.20.0Vd (V)
VG-VT=0.1V
VG-VT=0.4V
VG-VT=0.7V
VG-VT=1.0V
1.6
1.4
1.2
1.0
0.8
(eV)
SiNW(110) diameter:4nm temperature:300K
SiNW(100) diameter:4nm temperature:300K
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SCF CALCULATIONS AND BAND CALCULATIONS �
Self-consistent calculations
(exterior eigenproblem) •! Eigenproblems of large Hermitian
matrices called the Hamiltonians need to be solved self consistently
•! Self-consistent calculation –! calculate self-consistent charge
density
–! solve exterior eigenproblem
–! large number of eigenpairs
4
eigenvalue
Smallest
Au = λuA = AH ∈ Cn×n
u �= 0† Hasegawa, Y., Iwata, J.I., Tsuji, M., Takahashi, D., Oshiyama, A., Minami, K., Boku, T., Shoji, F., Uno, A., Kurokawa, M.,
Inoue, H., Miyoshi, I., Yokokawa, M.: First-principles calculations of electron states of a silicon nanowire with 100,000 atoms on
the K computer. In: Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and
Analysis. pp. 1:1–1:11. SC ’11, ACM, New York, NY, USA (2011)!
Band structure calculations
(interior eigenproblem) •! By using the self-consistent charge
density, various physical quantities can be calculated
•! The electronic band structure around the Fermi energy is important information on the electric current flow of the system
•! Band structure calculation –! solve interior eigenproblem
–! solve multiple eigenproblems with different parameters
–! relatively small number of eigenpairs
5
k (wave number)
eigenvalue
. . .
A(k)u = λu
Fermi energy
SCF calc.→ exterior eigenproblem � Band calc.→interior eigenproblem �
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二村保徳 筑波大 櫻井研
BAND STRUCTURE CALCULATIONS WITH
SAKURAI-SUGIURA METHOD �
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CONTOUR INTEGRATION �Contour integration
7
Re!
Im!
!.eigenvalue
"! Sk, V ∈ Cn×L
k = 0, 1, . . . ,M − 1
L,M << n
# of grid points
L arbitrary vectors (linearly independent)
Shifted Block CG 法�
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BAND STRUCTURE OF 10,000-ATOM SiNW �Band structure
38
0 0.1 0.2 0.3 0.4 0.5
0.06
0.07
0.08
0.09
0.1
0.11
0.12
k
eigenvalue
z
864x4 = 3456 nodes
for 12 hours
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BAND STRUCTURE OF 10,000-ATOM SiNW �
Band structure
39
0 0.1 0.2 0.3 0.4 0.5
0.096
0.098
0.1
0.102
0.104
0.106
k
eigenvalue
z
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内田和之 東大押山研 �
ATOMIC & ELECTRONIC STRUCTURES OF
TWISTED BILAYER GRAPHENE �
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!"#$%&'()*+++
%,+-
%,+-.
%,+/
%,+/.
%,+0
%,+0.
%1 %2 %0 %,3 %,- %31 %32 %30 %43 %4- %21 %22 %20 %.3 %.- %-1
5563
5763
8
8
8'&*
■ %#$
9
■ 8
8
▲ ▲9■ 8
1
8
8
!"#$%&'()*+++
55
57
■
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¡ コンパクトモデルを用いたSiNW-FETの電流電圧特性の計算 § ゲート電極の効果を第一原理的に扱う(産総研 大谷さん) § ソース•ドレイン電極も取り入れた第一原理輸送計算
¡ 櫻井 -杉浦法による大規模系のバンド計算(筑波大 二村保徳)
¡ 捻れ二層グラフェンの構造と電子状態(東大 内田和之) ¡ RSDFT-CPMDの実装(東大 小泉健一、阪大 重田照育)
§ Si 1000原子系、1ステップ 秒(20 Ry, 1024 node)
まとめ �