Computational Studies of Two-Dimensional

Materials: From Graphene to Few-Layer

Graphene and Beyond

Mei‐Yin ChouInstitute of Atomic and Molecular Sciences

Academia Sinicaand

School of Physics, Georgia Institute of Technology

Acknowledgment

The GT Graphene CenterWalt de HeerEd ConradPhil FirstZhigang JiangAndrew ZangwillMarkus Kindermann

Georgia TechJia-An YanSalvador Barraza-LopezWen-Ying RuanLede XianZhengfei Wang

Academia SinicaChing-Ming WeiChih-Piao ChuuYongmao CaiCheng-Rong Hsing

In condensed matter physics, we work with

“interesting” materials:

Fundamental physics 

AND a lot of possible applications

(e.g., energy applications – superconductivity, solar cells, energy storage, etc.)

The development of advanced materials is critical to the core challenges in renewable energy, 

electronics industry, defense, etc. 

http://www.whitehouse.gov/mgi June 2011

U.S. National Science and Technology Council 

• To emphasize the importance of materials research and the need to “discover, develop, manufacture, and deploy advanced materials in a more expeditious and economical way”.  

• The development of advanced materials can be accelerated through the use of computational capabilities in an integrated approach. 

U.S. National Science and Technology Council 

Interacting N electrons (as in condensed matter)

1 2 1 2

2 22

1 1

H ( , ,..., ) E ( , ,..., )

H ( ) ( )2 | |

N N

N N

i ext ii i i j i j

x x x x x x

eV xm x x= = ≠

Ψ = Ψ

= − ∇ + +−∑ ∑ ∑

v v v v v v

h vv v

Electrons are Fermions:

1 1( ,..., ,..., ,..., ) ( ,..., ,..., ,..., )i j jN i Nx x x x x x x xΨ = −Ψv v v v v v v v

Extremely Challenging for Theorists!

But N ~ 1023 !!

Fundamental Properties of Crystals

Al Si

No. of electrons 13 14

Type of material metal semiconductor

Crystal structure fcc diamond

nn distance (Å) 2.86 2.35

Cohesive energy (eV/atom) 2.39 4.63

Bulk modulus (Mbar) 0.72 0.99

First-principles studies

Evaluate the ground state energy using only the atomic numberas input

Standard Models for “First‐Principles” Calculations

• Input: Atomic numberAtomic arrangements

• Ground‐State Properties:

Density functional theory(“One‐particle” equation for the many‐body problem)

Quantum Monte Carlo method(Variational Monte Carlo and Diffusion Monte Carlo)

• Excited‐State Properties:

Many‐body perturbation theory(Quasiparticle spectrum, “GW” approximation)

AND these calculations provide input parameters for model studies.

Density Functional Theory

Hohenberg and Kohn, 1964; Kohn and Sham, 1965(1998 Nobel Prize in Chemistry)

For interacting electrons in an external potential, thetotal energy is a functional of the density, ( )n xv

23 3 3

0( ) ( )E [ ] [ ] ( ) ( )d [ ]

2 | | kinetic + potential + electrostatic + exchange-correlation

ext xce n x n xn T n V x n x x d xd x E n

x x′

′= + + +′−

=

∫ ∫∫v v

v vv v

[ ] : exact form unknownxcE n(Local approximation; Generalized gradient approximation)

Density Functional Theory (Continued)

Minimize with res E[ ] (pe Ect to : 0)n n xn

δδ

=v

One obtains a set of one-particle equations which can be solved self-consistently.

22 ( ) ( ) ( )

2 eff i i iV x x E xm

ψ ψ⎡ ⎤− ∇⎢ ⎥⎣ ⎦

+ =h v v v

2( ) | ( ) |N

ii

n x xψ=∑v v

2 3( ) [ ]( ) ( )| |

xceff ext

n x E nV x V x e d xx x n

δδ

′′= + +

′−∫v

v vv v

with

Charge density in Si (110 plane)

Pseudopotential Method

• Effective potentials felt by valence electrons due to the nucleus and the (frozen) core electrons

• Gives the same wave functions outside the core region, same total charge, and same eigenvalues

sp

d silicon

~ 1 / r

What can we get from these calculations?

From the total energy as a function of atomic positionsat T= 0 we get:

• phase stability (lowest-energy atomic configurations)• elastic properties• interatomic forces

...

Also electronic properties

Yin & Cohen, PRB 1982

The CDC 7600 was designed by Seymour Cray. The 7600 ran at 36.4 MHz (27.5 ns clock cycle) and had a 65 Kword primary memory using core and variable‐size (up to 512 Kword) secondary memory (depending on site). It could deliver about 10 MFLOPS on hand‐compiled code, with a peak of 36 MFLOPS. ‐‐Wikipedia

Control Data Corporation (CDC) 7600: 1971–1983

Finite-Temperature Thermodynamical Properties

Second derivative of the total energy with respect to atomic displacements

Interatomic force-constant tensors

Energy of phonons (lattice vibrations)

Entropy and free energy at T

, ,

( , )1( , ) ( ) ( , ) ln 1 exp{ }2

nn B

Bk n k n

k VF T V E V k V k Tk T

ωω⎡ ⎤

= + + − −⎢ ⎥⎣ ⎦

∑ ∑v v

vv h

h

Thermal Expansion Coefficient

Silicon

1 LL T

α Δ=

ΔWei and Chou,PRB 50, 14587 (1994).

40‐60 MHz, 64MB memory, 3x1.3GB disks, SunCD, SCSI

Sun Microsystems SPARC 2(Scalable Processor ARChitecture, USD $19K)

Two‐Dimensional Materials Created in the Laboratory

There exist many three‐dimensional layered materials: graphite, h‐BN, transition‐metal dichalcogenides, …

In the past few years, it has become possible to fabricate one layer of these materials and make measurements of their special properties in two dimensions.

National Energy Research Scientific Computing Center (NERSC)

“Hopper” ‐‐ 153,408 processor‐core Cray XE6 system1.05 petaflops

(Front)                                                      (Back Aisle)

2010 Physics Nobel Prize recognized the work on graphene

The development of this new material (graphene) opens new excitingpossibilities. It is the first crystalline 2D-material and it has uniqueproperties, which makes it interesting both for fundamental science and forfuture applications. The breakthrough was done by Geim, Novoselov, andtheir co-workers; it was their paper from 2004 which ignited thedevelopment. For this they are awarded the Nobel Prize in Physics 2010.– Royal Swedish Academy of Sciences

Few-layer graphene (FLG) prepared by mechanical exfoliation (repeated peeling).

“Using FLG, we demonstrate a metallic field-effect transistor in which the conducting channel can be switched between 2D electron and hole gases by changing the gate voltage.”

The Graphene Phenomenon

22,000 SCI papers on graphene published since 2004

.

Graphene

q1

q2

KM

K'

K

K

K' K'kx

ky

Γ

M K'

(Mark Wilson, Physics Today Jan 2006, p. 21)

X

(Wallace 1947)

VF ≈ 106 m/s ≈ c /300

Linear Dispersion

FE kυ= h

In a hypothetical, parity-noninvariant, gapless semiconductor the electron bands exhibit degeneracy points, i.e., points on the Fermi surface where the conduction and valence bands intersect. The low-energy electron dynamics are described by linearizing their spectrum about the degeneracy points and are thus modeled by relativistic Weyl fermions.

Dirac-Weyl Hamiltonian for massless relativistic fermions (e.g. neutrinos)

pseudospin = 1/2 (two sublattices)

Graphene: Relativistic Physics in a Nonrelativistic Material

relativistic gauge theories

κσ ⋅= ˆFvH h

κFvE h±=

Physical Review Letters

Electronic and Transport Properties of Graphene Systems

• Structural and electronic properties of oxidized graphenePRL 103, 086802 (2009)

• Effects of metallic contacts on electrontransport through graphenePRL 104, 076807 (2010)Nano Lett. 12, 3424 (2012)

• Landau levels in twisted bilayer grapheneNano Lett. 12, 3833 (2012)

• Anisotropic wave-packet dynamics and quantum oscillation in twisted bilayer grapheneNano Lett. (in press)

20nmX20nm

3.8nmX3.8nm 47nmX47nm

Miller et al. Science 324, 924 (2009)

Multilayer Epitaxial Graphene

Twisted bilayer graphene

What is “1” + “1” ?

× θ

Twisted Bilayer Graphene

Starting from AB‐stacked bilayer graphene, the bottom layer is fixed and the top layer is rotated.

Commensurate angle

Other θ values are incommensurate angles.

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

= −22

221

33cos

pqpqθ

Shallcross et al. Phys. Rev. B 81, 1 (2010)

Novel Properties of Twisted Bilayer Graphene

• Reduction of the Fermi velocity and von Hove singularities in the density of states

• Anisotropic transport in certain energy ranges

• Coupled Dirac fermions and neutrino-like oscillation

• Fractal-like energy spectra under magnetic field at small twist angles

Novel Properties of Twisted Bilayer Graphene

• Reduction of the Fermi velocity and von Hove singularities in the density of states

• Anisotropic transport in certain energy ranges

• Coupled Dirac fermions and neutrino-like oscillation

• Fractal-like energy spectra under magnetic field at small twist angles

Free Electrons in a magnetic field (Landau levels)

quantization of cyclotron orbits BEn

ergy

Magnetic field

En =(n+ 1

2)heB

m*

2( )E k k∝

En =± 2ehc2Bn

Magnetic field

Ener

gy

EF

( )E k k∝

(0.1 meV per Tesla)B=1 T

E(n=1) = 40 meV

Free Electrons in a magnetic field (Landau levels)

quantization of cyclotron orbits BEn

ergy

Magnetic field

En =(n+ 1

2)heB

m*

2( )E k k∝

En =± 2ehc2Bn

Magnetic field

Ener

gy

EF

( )E k k∝

(0.1 meV per Tesla)B=1 T

E(n=1) = 40 meV

No Lattice!!

Hofstadter Butterfly (Hofstadter, PRB 14, 2239, 1976)

• A rare occurrence of a nice fractal‐like picture in quantum mechanics• 2D electrons with a periodic potential in the presence of a strong 

magnetic field

Two co‐existing length scales:  

magnetic length and lattice constant

φ = magnetic flux through one unit cell

φ/φ0

E

Hofstadter Butterfly (Hofstadter, PRB 14, 2239, 1976)

• A rare occurrence of a nice fractal‐like picture in quantum mechanics• 2D electrons with a periodic potential in the presence of a strong 

magnetic field

The energy bands are clustered into subgroups and subcells; spectra of almost Mathieu operator; self‐similarity maps; gaps are labeled using a Diophantine equation with parameters related to Chern numbers.

A possibility to measure this spectrum on ultra cold neutral atomsD. Jaksch and Peter Zoller, New Journal of Physics 2003

φ = magnetic flux through one unit cell

φ/φ0

E

Nemec and Cuniberti, PRB 74, 165411 (2006)

φ/φ0B = 40 T →  φ/φ0 = 0.001φ = magnetic flux through one unit cell

20nmX20nm

3.8nmX3.8nm 47nmX47nm

Miller et al. Science 324, 924 (2009)

Multilayer Epitaxial Graphene

Twisted bilayer graphene

Tight Binding Hamiltonian 

TB parameters are obtained by fitting the TB bands to reproduce the band structure obtained from first‐principles calculations

θ=3.89O

Real space Hamiltonian (Hermitian matrix)

Construct a new orthogonal basis

ji

j

iji

ij aaldAietH +∫∑ ⋅= )exp(,

vv

h

NNNNNN

NNNNNNNN

bHa

baHb

ΦΦ=ΦΦ=

Φ−Φ−Φ=Φ=Φ −+++

~~

~1111

Hamiltonian in the new basis⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

OMMM

L

L

L

22

211

10

0

0

abbab

ba

H

Real space Green’s function of the first element

(continued fraction expansion)L

L−−+−−+

−−+=ΦΦ

2

22

1

21

0

001)(

aiEbaiE

baiEEGr

ηη

η

Lanczos Recursive Method

140nm×140nm, over 1.5million atoms

Landau Levels at Small Twist AnglesCommensurate:

θ=2.56292Oθ=1.64996Oθ=1.06689O

θ=2.1Oθ=1.5Oθ=1.2OIncommensurate:

Summary

• Twisted multilayer graphene exhibits intriguing electronic properties.

• A complex Hofstadter butterfly spectra could be observed in twisted graphene bilayer within a certain angular range at laboratory accessible magnitudes of magnetic field.

• In addition to the periodicity of the supercell, the interlayer coupling also plays a role in producing the fractal‐like spectra.

Novel Properties of Twisted Bilayer Graphene

• Reduction of the Fermi velocity and von Hove singularities in the density of states

• Anisotropic transport in certain energy ranges

• Coupled Dirac fermions and neutrino-like oscillation

• Fractal-like energy spectra under magnetic field at small twist angles