Novel Quantum States in Condensed Matter (NQS2022)
Kyoto Univ, Nov. 1-4, 2022

Physics of Graphene&moiré materials

Mikito Koshino (Osaka University)Graphene (2004): the thinnest material ever created・Honeycomb lattice ofcarbon atoms
・1 atom layer thick!

Grapheneʼs properties
・Thin
Only 0.3nm thick・Light・Stiff
1m2 perfect graphene can support 4kg
・Cheap!
Carbon everywhere in the world・Highly conductive
Electron speed at room temperature is
100 times larger than in Si.

Application
Flexible electronics

Wearable technologies

University of Manchester

http://mobile.technologyreview.com/Super-capacitorsSensors

University of Manchester

Sensors 2020, 20(5), 1506

Graphene from graphite
Graphite : 3D stack of graphenes
Intralayer : covalent bonds … StrongInterlayer : van der Waals force … Weak
Easy to peal off
Graphite in pencil leads

Presented by Dr. K. Tsukagoshi (NIMS)

Writing in pencil
Make graphene flakes
by rubbing the tip with paper

How to make graphene?
What you need:

Scotch tape

Mono-crystalline graphite(Zoom in)Presented by Dr. R. Nouchi (Tohoku)

How to make graphene?
scotch tapegraphite
fragmentpushpeel off(repeat)
scotch tape with graphite

Rub onto the substrate(scratch with a plastic stick)

Substrate(SiO2)(by the courtesy of Maika Takita in Columbia U)

How to make graphene?
Search using optical microscope!

1-layer

1-layer2-layer(by the courtesy of Maika Takita in Columbia U)

2-layer

Discovery of graphene (2004)

A. Geim, K. Novoselov(Nobel prize 2010)

Graphene to 2D materials
hexagonal BNGraphene

Transition metal dicharcogenides

MoS2

Phosphorene

Graphene is not just a thin metal

Electron’s graphene= “Relativistic massless fermion”

Carbon atom and hybrid orbitals
Carbon atom

4 valence electrons𝒔

𝒔𝒑𝟑𝒑𝒙𝒑𝒚𝒑𝒛

𝒔𝒑𝟐

http://www.homesweetlearning.com/

Diamond vs graphite
Graphite

Diamond

𝒔𝒑𝟐 bonding

𝒔𝒑𝟑 bonding… 𝒑𝒛 electrons do notparticipate in bonding

𝒔
𝒔𝒑𝟑

𝒑𝒙

𝒔𝒑𝟐

𝒑𝒚𝒑𝒛
Free electrons

Band structure of graphene

𝒑𝒙𝒔
𝒔𝒑𝟐

𝐸#𝒑𝒚𝒑𝒛𝒑𝒛

Band structure of graphene
𝐸𝒑𝒛 orbital
𝑝𝑦𝑝𝑥

SiliconGraphene

Equivalent tomassless Dirac fermion

Dirac fermion
Relativistic

Classical𝑝%
𝐸=2𝑚: parabolic

𝐸 % = 𝑐 % 𝑝% + 𝑚𝑐 % % : hyperbolic
EE

𝐸 = 𝑚𝑐 %p（gradient ±𝑐）pnever exceeds
light speed

Massless Dirac fermion
Reduce 𝑚 in

𝐸 % = 𝑐 % 𝑝% + 𝑚𝑐 % %𝑚=0EEE𝑚𝑐 %p𝑚𝑐 %pp
𝐸 = ±𝑐𝑝

Grapheneʼs electrons cannot stop!
Conventional particle

velocity :Energy E(E-p gradient)momentum p

Graphene

energy E

velocity :… independent of p!!
(~ 1/300 of light speed)v

momentum pv
v

cannot stopKlein tunneling
Conventional particles
Cannot overcome potentialbarrier
higher than the kinetic energy

Graphene
Electron can overcome however high barrierowing to the negative energy states!

Klein tunnelingTwisted bilayer graphene (TBG)
Rotational stack of two graphene layers

moiré patternMoiré period

𝜃 = 2∘

𝜃 = 5∘

L = 7.04nm

2.82nm𝜃 = 10∘1.41nmEnergy bands of TBG

Graphene’s band:

Energy (meV)”Dirac cone”(linear band)

Flat bands appear
at a “magic angle”

R. Bistritzer and A. MacDonald, PNAS 108, 12233 (2011).
E. Morell, J. Correa, P. Vargas, M. Pacheco, and Z. Barticevic, Phys. Rev. B 82, 121407 (2010).
G. Trambly de Laissardiere, D. Mayou, and L. Magaud, Nano Lett. 10, 804 (2010).
S. Shallcross, S. Sharma, E. Kandelaki, and O. Pankratov, Phys. Rev. B 81, 165105 (2010).

Y. Cao et al, Nature 556, 43 (2018)

Y. Cao et al, Nature 556, 80 (2018)

Superconductivity is observed in the magic angle TBG!

Family of moiré 2D materials
Twisted bilayer graphene

Twisted multilayer graphenes

Park et al, Nature 590, 249 (2021)

Graphene-hBN

Dean, et al, Nature 497, 598 (2013)

ABC-trilayer graphene-hBN

TMDC-TMDC
C. Jin et al,Nat. Phys. 15, 1140 (2019)

Chen et al Nature 572, 215 (2019)

Lecture plan
1. Graphene (day 1)
Dirac electrons in graphene
Pseudospins & topology
Landau levels
Singular orbital diamagnetism

2. Moiré 2D materials (day 2 & 3)
Twisted bilayer graphene
Flat bands & superconductivity
Continuum model
Hofstadter butterfly
Twisted quasicrystals

Derive graphene’s Dirac equation

Gaphene’s Dirac equation

EE
pypx

pypx

Warm up: 1D tight-binding modelGraphene𝒔𝒑𝒙
𝒔𝒑𝟐

𝒑𝒚𝒑𝒛
𝒑𝒛

EF

Graphene

(From the previous page)(From the previous page)

rewrite 𝒌 − 𝑲 → 𝒌rewrite 𝒌 − 𝑲′ → 𝒌Graphene’s Dirac equation

Why band touching at K?
Ex). Space inversion: P
When [H,P]=0 (inversion symmetric),
an eigenstate of H is also an eigenstate of P.
Eigenvalues of P = 1, -1 (as P2 = 1)
P = -1 (odd)

P =1 (even)×
Never hybridized

Why band touching at K?

K, Kʼ points:
Invariant in 120 rotationK’

120 degree rotation: C3
When [H, C3]=0 (120 rotation symmetric)
an eigenstate of H is also an eigenstate of C3.

KK’K

Eigenvalues of C3 ：1, w, w2[∵since (C3)3 = 1]
C3=1

C3 = w

C3 = w 2

Different sectors never hybridized

K’K

Why band touching at K?

=1

==

𝜔

|𝐾, 𝐵⟩|𝐾, 𝐴⟩

120∘ rotation

𝑪𝟑 = 𝝎∗120∘ rotation

×𝑪𝟑 = 𝝎Never hybridizedà degenerate at Dirac point𝜔∗Dirac equation (relativistic quantum mechanics)

Dirac equation
4x4 matrix form:

positive energy, spin↑
positive energy, spin↓
negative energy, spin↑
negative energy, spin↓(near p=0)

Massless limit
Reduce 𝑚 in

𝐸 % = 𝑐 % 𝑝% + 𝑚𝑐 % %𝑚=0EEE𝑚𝑐 %p𝑚𝑐 %pp
𝐸 = ±𝑐𝑝Massless limit
0Unitary transform

1,2 component (L-handed)or

3,4 component (R-handed)

Weyl equationorWeyl Hamiltonian

Graphene

Left-handed

near K valley

Right-handed

near Kʼ valleyz 成分がない（2次元なので）velocity

velocity

wavefunction

wavefunction
spin ↑spin ↓A-site
B-site… Pseudospin

Graphene’s pseudospin & band topology

Wavefunction of graphene’s electron
Schrödinger equation
expectation value of pseudospin ⟨𝝈⟩=𝑣𝝈 ⋅ 𝒑𝒑𝝈 ⋅ " = +1

Eigenenergy and eigenstate
pyp
qpx𝒑

AB sublattice = “pseudospin”

𝝈 ⋅ " = −1

Pseudospin in graphene

Pseudospin always
on a single (xy) planePT-symmetry!!
P : space inversion
T : time reversal

Pseudo-spin restriction in PT-symmetry

Applicable to any 2x2 PT-symmetric
Hamiltonian without SO couplingSpins on 2D plane à Winding number can be defined
Any closed path C is classified by the winding number(= How many times the spin rotates when going around C)
90°

180°

0°

270°

0°

90° 180° 270° 360°

winding number = +1

0°

90° 180° 270° 360°

winding number = 00°

90° 180° 270° 360°

winding number = -1

Pseudospin in graphene

𝜉=+winding number = +𝟏

𝜉=−winding number = −𝟏

Nonzero winding number = band touching point

𝑪Nonzero winding number on a path C

Band touching (a singular point)
must exist inside C

Band touching point is robust
A band touching point never disappears
by a continuous deformation under PT-symmetryBand touching points with opposite winding numbers
can pair-annihilate+1-1

winding number = 0

Continuousdeformation

winding number = 0

Pair-annihilation of Dirac points in graphene
G. Montambaux, C. R. Physique 19, 285 (2018)

tt’

Bilayer graphene (BLG)Stacking configuration
AB-stacking (natural graphite)

StableAA-stacking

UnstableBand structure of AB-stacked BLG
Effective Hamiltonian
0.334nmInterlayer
coupling
g1 ~ 0.4eV

Energy bands

Quadratic band touch
McCann and Fal’ko, PRL 96, 086805 (2006).

2x2 effective Hamiltonian
4x4 effective Hamiltonian

2x2 effective Hamiltonian (near E ~ 0)𝐴!

𝐵"

Pseudospin in AB-bilayer graphene𝐴!

𝐵"

2x2 effective Hamiltonian𝜉=+winding number = +2

𝜉=−winding number = -2
Band touching protected by PT-symmetry

Field-tunable energy gap
E-field
layer 1layer 2+D-D(PT-symmetry broken)+D-DValley Hall conductivity
Berry curvature
K’K
E-field
layer 1layer 2+D-D𝐸𝐹*𝜎()= −𝑒 + /ℎ!

* = +𝑒 + /ℎ𝜎()

Y. Shimazaki, et al, Nat. Phys. 11, 1032 (2015)

Absence of back-scattering
Hamiltonian:V(r) : Disorder potential (scalar)
Direction of pseudo spinV(r) does not flip
pseudo-spin

Cf. Absence of localization in metallic carbon nanotube
Ando and Nakanishi, JPSJ. 67, 1704 (1998)
Ando, Nakanishi, and Saito, ibid. 67, 2857 (1998)

Localization in graphene

Nomura, Koshino, Ryu, PRL 99 (2007)

Conductivity (Thouless number)
Hamiltonian:
SCBA

Disorder potential

g (in log scale)“Symplectic”

L =6,8,10,12,14EF

Fermi Energy

Conductivity increases
as L increases

Pseudo-spin and back-scattering
Monolayerabsence ofback-scatteringanti-localization

Bilayerpresence ofback-scattering

localization

Localization in bilayer graphene

Koshino, PRB 78, 155411 (2008)

Bilayer

Monolayer
SCBA
g (in log scale)g (in log scale)

SCBA

Fermi Energy
Conductivity increases as L increases(anti-localziation)

Fermi Energy
Conductivity decays as L increases(localziation)
Cf. Weak localization of bilayer:
Gorbachev et al, PRL 98, 176805 (2007)
Kechedzhi et al, PRL 98, 176806 (2007)

Graphene in magnetic fields

Magnetic field effect on graphene
Graphene

Conventional 2D electron
B-eevB2,+1,+v

432

n =0, 1, 2, …

0zero-energy Landau level

11,-02, -McClure, Phys. Rev. 104, 666 (1956).

Conventional 2D electron

GrapheneLandau levels of conventional 2D electron
Hamiltonian
where
do not commute!

Ladder operatorsthen43210

Landau levels of graphene
Hamitonian（Ladder operators）

where

2,+1,+0
zero-energy Landau level1,-∴

n =0, 1, 2, …

2, -Diamagnetismc (x 10-5)
Diamagnetism= property of an object to repel a magnet

Levitating frog (water)

Water
Copper
DiamondSilverBismuthGraphite (HOPG)
Superconductor

-0.91
-1.0
-2.1
-2.6
-16.6
-40.0
-105

16T

Graphite (HOPG)0.6T

Geim (2000)
sample by Dr. Nouchi (Tohoku)
HOPG＝ Highly Oriented Pyrolytic Graphite

Superconductor

Magnetic levitation of graphite
From “InnovativeMaterials”

Magnetic levitation of graphite
Pencil leads

Top view

NS

NS

N

S

Neodymium magnet（0.6 Tesla on surface）

S

N

S

N

S

N

S

NS

Diamagnetism of graphene

McClure, Phys. Rev. 104, 666 (1956)
Koshino and Ando, Phys. Rev. B 76,085425 (2007)

Graphene: Strong diamagnetism at Dirac point
Fermi energy
Susceptibility -c

Dirac point

low Thigh T

T→0:

Singular diamagnetism
at Dirac point

Optical motion control of maglev graphite
M. Kobayashi and J. Abe,J. Am. Chem. Soc.134, 20593 (2012)

Why following light spot?
Fermi energy

Susceptibility |c|

Dirac point

low T : higher diamag
high T : smaller diamag
Forcehigh T→ lower energy cost in B-field→ graphite attracted to high T area!

Why diamagnetism diverges?
Landau level at E=0 → Total energy U definitely increases in B
→ Magnetization
B=0

B>0
Landau level spacing：×Number of electrons
per Landau level：＝Total energy increase：

Magnetization
Susceptibility
divergesis negative!

Monolayer vs BilayerBand structure

Monolayer graphene

SusceptibilityeF

eF-c(eF)

Bilayer graphene

eF
g1( |eF|< g1 )-g1-c(eF)

Log peak (weak singularity)
Safran, PRB 30, 421 (1984)
Koshino, PRB 76, 085425 (2007)

wave number kx(units of -2e2v2/pg1)

Hall effect
Hall conductivity

Edwin Hall

Electrons

Holes

Electron / hole can be
distinguished by sign of 𝜎()Quantum Hall effect (1980)
𝜌() (= 1/𝜎() )Klaus von Klitzing

𝐵Hall conductivity accurately quantized in 2D electron gas𝑒%𝜎'( = − × 𝑁,ℎ𝑒%1=ℎ25812.8074555Ω

http://padenroder.com

Conventional 2D

Graphene