新学術領域「コンピューティクスによる物質デザイン：複合相関と非平衡ダイナミクス」
計画研究「A02-4：ナノ構造形成・新機能発現における電子論ダイナミクス」

Twisted Bilayer Graphene のRSDFT コードを用いた大規模電子状態計算
東京大学工学系研究科物理工学専攻

内田和之、古家真之介、岩田潤一、押山淳

twist angle: θRSDFT code

Good collaboration ofcomputer scientists &physicists■A real-space DFT code
J.-I. Iwata, D. Takahashi, A. Oshiyama, T. Boku,K. Shiraishi, S. Okada, and K. Yabana, J. Comp. Phys. 229, 2339 (2010).

■LDA , GGA with spin-polarization
S. Furuya…

■Norm-conserving pseudo-potential⇒2011 Gordon-Bell Prize■Car-Parrinello Meta-Dynamics

Silicon nano-wire (10000 Si atoms),
24 hours to get SCF using K computer
Recent Developments

Y. Shigeta(This talk)■Ultra-soft pseudo-potential
K. Uchida

Electronic Structure Calculation
for twisted Bilayer Graphene(carbon)

Ultra-soft pseudopotential Method
・Pseudo-potential calculation⇔

only valence electrons, low cost{・All-electron calculation
valence electrons + core electrons

Kohn-Sham equation■Norm-conserving pseudopotential{T + Vlocion+ VHXC + Σi|βi>Di<βi|}|Φk> = εk |Φk>Soft for Si, Al, ...
×■Ultra-soft pseudopotential (This Work){T + Vlocion+ VHXC + Σij|βi>Dij<βij|}|Φk> = εk S|Φk>
Generalized eigen-value problem

Soft for H,C,N,O,Ca, Ti, Fe, Cu,......
Many important elements
OKLow cut-off energy (→less calculation cost, less memory)

Points in Implementation
Kohn-Sham equation{■Norm-conserving pseudopotential{T + Vlocion+ VHXC + Σi|βi>Di<βi|}|Φk> = εk |Φk>■Ultra-soft pseudopotential (implemented in This Work){T + Vlocion+ VHXC + Σij|βi>Dij<βij|}|Φk> = εk S|Φk>

Generalized eigen-value problem■Inner product:{}<Φm|S|Φn>=<Φm| 1+Σij|βi> qij <βi| |Φn>■Charge density:

ρ(r) = Σl Φl*(r)Φl (r) + Σl Σij <Φl|βi>Qij(r)<βj|Φl>■update of non-local potential

Dij=Dij(0) + ∫ Vloc(r)Qij(r)drQij(r) : localized function
around an atom{“<βi|Φm>” must be calculated
many times.
→ time-consuming partIn real-space formalism,

・“Egg-box effect”(innaccuracy in integrations)・To avoid this, we use
Ono-Hirose integration method
or mask-function method.

What is Twisted Bilayer Graphene (tBLG)?
■ θ=0° : AA-stacking

θSide view:

4 atom/cell

1st graphene layer■ θ=60° : AB-stacking

2nd graphene layerZ

4 atom/cellXYTop view:,60°)■ θ=any value(≠0°(e.g.) θ=6°

θ=20°

twist angle: θYZX{

“Moire pattern”

・periodic only for special θs (→this work)
・non-periodic for other θsHow to make tBLG?
(e.g.) Epitaxial growth of graphene layers on SiC (0001) ( = carbon surface)
Often Twisted to each otherJ. Hass et al.,PRL 100,125504 (2008)

carbon atom positionbright region (many atoms)■ Superstructures in STM image← Non-AB, Non-AA stacking of layers(Moire pattern due to twisted layers)■ Periodicity → get twist angle θ(c.f.) C-C~1.42Å
in a graphene layer

STM images■ Many patterns (many θs)Why tBLG?
①STM+STS for Twisted Graphene Layers

1.0“Fermi-velocityreduction”

0.6
0.4
0.2?

05

10 15 20 25
twist angle θ (°)▲ : STM+STS: continuum tight-binding calc.
★ : DFT calc(■consistent with tight-binding calc. for tBLGs
30[Trambly et al., Nano Lett 10, 804 (2010)]

THIS WORK :STS under magnetic field→Landau levels→Fermi velocity：VF■Fermi velocity VF is controlled by changing
twist angle θ.

★

0.8(STM→ Moire → twist angle：θ

However,controversial■ ②ARPES
SLG

Fermi velocity vF / vF

SLG

Luican et al. : PRL 106, 126802 (2011)

Fermi velocity vF / vF■

⇔Hicks et al. : PRB 83, 205403 (2011)
1.0“No Reduction”

0.80.6

Continuum tight-binding calc.

0.40.200

ARPES data

△5

Full tight-binding calc.

1015twist angle θ (°)

We wish to know if the Fermi-velocity reduction is true or not.
→ DFT calculations for tBLGs

20

Model & Method■Fermi-velocity reduction is prominent when θ< 2 ° [STM/STS by Luican]→ We calculate a periodic tBLG with θ=1.89 °unit cell (θ=1.89°)

1st layer=overlapped by

2nd layer+■3676 atoms/cell when θ=1.89 ° (extremely large system)→ RSDFT with Ultra-Soft Pseudo-Potential＊In tBLG, θ is small (# of atoms is large) → quantum effect is strong
Details:■LDA (PZ81)■Inter-atom distance in a layer = fixed at 1.42Å■Real-space mesh interval=0.23Å( ~50Ry in plane-wave method) ■Inter-layer distane = fixed at 3.4Å
■Vacuum width to separate a tBLG and images = 9.6Å
■k-point sampled: Γ point

Band Structure & Fermi-Velocity reduction
θ=1.89°0.2
Dirac point0(eV)Dirac cone of tBLG

-0.2
-0.4

energyFermi level=0eVK

Brillouin zone of tBLG

bands of tBLG
Dirac cone of SLGΓ

Fermi velocity VF/VFSLG

10.4

0.8
0.6
0.4
0.2
002k-space

68104
twist angle θ(°)

Fermi velocity= gradient of Dirac cone

DFT calculations predict thatthe Fermi velocity is reduced when θ < ~10°……support STM/STS experiment1st graphene layer
2nd graphene layer

AAAA
AB~75 A

θ=1.89°BAAA~75 AAA

Locally “AA-stacked” region, and “AB-stacked” region exist in the unit cell

Wave functions at the Fermi level
4 degeneratestatesABAAYXZ

4 wave functionsat the Fermi level

321arb. unit

Wave functions at the Fermi level are localized at the AA-stacked region.
……can be observed by STM

Moire pattern
θ=5°

AB

θ=10°small θlarge θ
θ=20°

θ=30°

⇔

AA

qualitatively
different

Triangular Lattice consisting ofAA- and AB-staking regions.
(only when θ < ~10°)⇒⇒
Wave functions at the Fermi level
are localized at AA-stacking regions

Wave functions at the Fermi levelare delocalized

Moire pattern in the atomic structure is reflectred on electronic structure.

Summary■ Large-scale DFT calculations fortBLGs with the twist angle θ ≧1.89 °■ Only when θ ＜ ~10°,{Electronic Structure• Fermi-velocity reduction• Localization of wave functions at the AA-stacked region→ Support STM/STS experiment■ Only when θ ＜ ~10°,

Atomic Structure• Moire pattern makes a triangular lattice consisting ofAA and AB stacked regions.

①θ→+0AAAB②θ=0AAAA- and AB-stacked regionsin the unit cellAA⇔

Only AA-stacked region
in the unit cell

■If coherent length of electrons is longer than the unit-cell size of ①,① and ② are different for the electrons.
■Otherwise, electrons feel that AA region in ① and ② are the same.

Periodic tBLG
1st layer graphene(N,M) tBLG(-M,N+M)

60degree Lhexagonal
super cell

(N,M)

(0,1)
(1,0)
primitive cell

2nd SLGhorizontal

L +θ/2

1st SLG+θ/2
rotationsuper cell-->band folding

band structure (k-space)is also rotated by +θ/2

θ

mirror symmetry in atomic arrangements

2nd layer graphene(-N,M+N)-θ/2
rotation

LIntra- and Inter- layer mixing of Bloch waves.

(M,N)

L

■Electronic structure of tBLG
= (folding)+(rotation)+(mixing) of two SLGs

-θ/2

■Moire pattern in Atomic structure

60degreesuper cell-->band folding

band structure (k-space)is also rotated by -θ/2→?Mioire pattern? in electronic structure?

Periodic tBLG(-M,N+M)

60degree L+θ/2

horizontal

L
(N,M)

+θ/2
rotation

primitive cell

■ relation between θand (N,M)
cosθ=

N2+4NM+M22(N2+NM+M2)■ # of atoms in the unit cell of BLG
Natom=4(N2+NM+M2)(e.g)
・(N,N) BLG = AA stacking BLG
θ=0 degree
・(N,0) BLG = AB stacking BLG
θ=60 degree
・(7,6) BLG : Natom=508
θ=5.08 degree
・(18,17) BLG : Natom=3676

Natom is very large when θ is small.

θ=1.89 degreereal spaceenergyBrillouin zone of tBLG

Dirac cone
Fermi velocity

K_origk-spacek-pace① folding

SLG has Dirac cone

② rotation

at K_orig and K’ _orig

1st SLG① supercell② rotation

BZ (primitive) Korig.①

BZ (primitive,rotated)

K’ orig.①

rotation(+θ/2)

BZ (folded, rotated)

K’K

foldoverlap
Korig.① Korig.②

BZ(folded, rotated)=Brillouin zone of tBLG

K’

2nd SLGK

BZ (primitive) Korig.②

BZ (primitive,rotated)

K’ orig.②

rotation(-θ/2)

BZ (folded, rotated)

K’K

foldoverlap{Korig.① and Korig.② in the BZ (primitive)
are folded to
K and K’ in the BZ (folded, rotated) when (N-M) mod 3 =1,2
G in the BZ (folded, rotated)when (N-M) mod 3 =0tBLG with θ=1.89 degree (N,M)=(18,17)→,(N-M) mod 3 =1
→Dirac cones in the primitive BZ are floded to K and K’1st layer graphene (SLG)primitive cellsuper celllattice vectors

lattice vectors(-M,N+M)

60degreeT2{

reciprocal lattice vectors

T1(N,M)

A2A1{A1A2A3{B1B2B3

T1 = NA1+MA2
T2 = -MA1+(N+M)A2
T3 = A3

reciprocal lattice vectors{S1S2S3

primitive cell

xS1+yS2 = αB1+βB2
T1(xS1+yS2) = T1(αB1+βB2 )∴2π x = (NA1+MA2)(αB1+βB2 )= 2π (Nα+Mβ )x = Nα+Mβ
2π y = {-MA1+(N+M)A2}(αB1+βB2 )= 2π {-Mα+(N+M)β }y = -Mα+(N+M)β(e.g) Korig. : (α,β)=(2/3,1/3) ----> K’ : (x,y)=(1/3,2/3)
when (N-M) mod 3 = 1(j=1)
M-N=3i+jx=(2/3)N+(1/3)M=(2/3)N+(1/3)(N+3i+j),

i : integer, j=0,1,2=(N+i)+(1/3)jy=(-2/3)M+(1/3)(N+M)=(-2/3)(N+3i+j)+(1/3)(2N+3i+j)=-i-(1/3)jKorig.

K’