unraveling the secrets of graphene by multiscale simulations

Accelrys Contract Research Services

Unraveling the secrets of graphene by multiscalesimulations

07102010

Dr. Johan Carlsson

Contract Research

Contract Research Services

• The Nobel Prize in Physics 2010 was awarded jointly to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene"

Nobel prize in Physics 2010

Andre Geim Konstantin Novoselov


• Graphite is the thermodynamically stable form of pure carbon.

• Graphene is the individual layered building blocks of graphite.

Graphene, the new old material


Carbon in different dimensions

0-D 1-D 2-D 3-D

FullerenesNanotubes

Graphite

Diamond

Graphene

Graphene is the mother of sp2-carbon and is now filling the missing link

from 0- 3 dimensions.


• Transistors

• Sensors

• Solar cells

• Trans-electrode membrane

Applications of graphene

Lin et al., Science 327, 662 (2010).Schedin et al., Nature Materials 6, 652 (2007).

Xang and Müllen, Nano Lett. 8, 323 (2008).

Garaj et al., Nature 467, 190 (2010).


Scientific interest in graphene

• The first true 2D-crystal

• The electrons in graphene behave as massless relativistic particles.

Relativistic space-time cone


• Graphene: The material for next generation of nanoelectronics?

• Simulations of fundamental aspects for device engineering:– Electronic structure of single layer graphene– Multilayer effects– Strain engineering– Edge effects in graphene channels– Role of defects?– Functionalization

• Scientific interest in graphene– Can a 2D crystal exist?– Are the electrons in graphene relativistic?

Outline


• The electrons in Graphene has very high mobility and long scattering length• Graphene would form extremely thin conduction channels

Graphene in nanoelectronics

Josephson junction with graphene film as

normal conductor

Heersche et al., Nature 441(2007)

Trauzettel et al., Nature Physics 3 (2007)

Spin qubits in graphene quantum dots

Lin et al., Science 327, 662 (2010).

High electron mobility transistor (HEMT)


• Exfoliation of graphite

• Oxidation of graphite

• Heating of SiC

• Epitaxial CVD growth

Synthesis of graphene

Forbeaux, et al, Phys. Rev. B 58, 16396 (1998).

Mattausch et al., PRL 99, 076802 (2007).


K.S. Novoselov, PNAS 102, 10451(2005); and Science 306, 666(2004)

Studying graphene with pen and paper


Electronic structure of Graphene

Unit cell with two atoms

Assume that the -band is a linear

combination of pz-orbitals centered at A

and B lattice sites

=C1 A+C2 B

Insert into the Schrödinger Equation:

H =E

And multiply with the pz-orbitals.

A B

Tight binding description after Wallace, Phys. Rev. 71, 622 (1947).


Tight binding for Graphene

This gives a two dimensional secular equation

With the solution

Summing over the nearest neighbors B gives the

matrix element H12

Summing over the next nearest neighbors A’ gives

the matrix element H11

02

1

2221

1211

C

C

ESHH

HESH

1211

2

12

2

22112211 4)(2

1HHHHHHH

SE

2cos4

2cos

2

3cos41 2

0

2

12

akakakH

yyx

A B

A’A’

A’

A’A’

A’

2cos

2

3cos2cos2011

akakakEH

yxy


• The Fermi surface is limited to the six corner points in the Brillouinzone. The band structure has linear dispersion around these Dirac Points.

Tight binding band structure for Graphene

Tight binding for Graphene

K

M

Brillouin Zone

kx (Å-1)


Preparation:

• Enter a graphene sheet into a supercell

Ab-initio calculations

•DFT calculations:

•Castep: Plane waves, Energy cut-off: 450 eV

•DMol: Localized orbitals, Radial cut-off 12 au

•Exchange-Correlation: PBE-GGA

•Atomic relaxation until Force/atom < 0.01eV/Å


• Graphene has two types of

bands: and

• Graphene is a zero gap semi

conductor with linear dispersion

at the Dirac point.

DFT band structure of graphene

M

K


Measurements of the electronic structure

Photoemission spectra of graphene grown on SiC

T. Ohta et al., PRL 98, 206802 (2007)

Conductivity of exfoliated graphene.

Novoselov et al., Nature 438, 197 (2005)

K M

-F

(eV

)


• Weak interactions between the layers in few layer graphene gives subtle effects on the electronic structure.

Layer effects on the electronic structure


Bilayer of graphene

1L Graphene 1x1 2L Graphene 1x1

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

The vertical interaction between the two layers split the -band.


Multi layers of graphene

1L Graphene 1x1 2L Graphene 1x1

4L Graphene 1x1 Graphite 1x1


• The graphene sheet may buckle under compression

• The bonds are extended under strain

Strain Engineering of Graphene

4% compression

of lattice const


• The bonds are shorter under compression leading to increased dispersion and broader band width.

• The bonds are elongated under strain leading decreasing dispersion and narrower band width.

• No gap opening under moderate homogeneous strain, since no symmetry breaking.

Electronic effects of strain

-4% compression

- 4% stretching


• The electronic structure is affected if the graphene sheet is cut into ribbons to make conduction channels in field effect transistors.

Edge effects in graphene channels

Wang et al., Phys. Rev. Lett. 100, 206803 (2008)


Electronic structure of graphene ribbons

The band structure is similar to a nanotube:

The wave functions in the graphene ribbon have dispersion along the ribbon, but they are quantized perpendicular to the ribbon. But …

Quantization


Bandstructure of the zig-zag edge

There are two types of edge states at a zigzag edge:

Localized sp2-dangling bond state

Semi localized pz-state

QuantizationSTM picture of a zigzag edge on graphite.

Giunta and Kelty, J. Chem. Phys. 114, 1807 (2001).


Bandstructure of the zig-zag edge

The anti ferromagnetic magnetic state is most favorable.

Bare edge, FerromagneticBare edge, Anti ferromagnetic

Son et al., Nature 444, 347 (2006).


Vacancies in graphene


• Vacancies take the form of a combination of a large ring plus pentagons. • No spontaneous defect formation in equilibrium in graphene.

Vacancies in graphene

odd motifs even motifs

JMC and Scheffler, Phys. Rev. Lett 96, 046806 (2006)

V3 V4

V1 V2

Eform( C ) = Emotifslab(NC) + NC C -EG

C = C - ½ EC2 (eV)

Efo

rm(e

V)

D3

SW

V1

V3

V2

V4V5V6

Formation energy

G free C-atom

at T = 0 K


• Heat of formation for defective graphene is comparable to other metastable forms of carbon such as nanotubes.

• Defects may be formed during growth and frozen into the structure.

Graphene sheets with moderate amount of defects

Heat of formationj

form

j

jCnGVG ECHHi

ii1

1


• The defects induces a pz-state and a dangling bond at zigzag atoms

Electronic structure of defects in graphene

Density of States (DOS)-point wave function

(isosurface for 0.1 e/Å1.5)

+ --

db-state for V1

+

+

--

JMC et al., PRL 96, 046806 (2006)

-SW

–V1

–V2

+- +

+

pz-state for V1

pz-state for V2


Chemistry of graphene

Böttcher et al., Nanotech. 17, 5889 (2006).

Functionalization Burning


Ab-initio Thermodynamics

E form(T ,p) {EVNC (NO ) Fvib(T ) NC C} {EG NO O(T ,p)} q F

),(),(),( pTNpTGpTG i

i

iform

Gas(T ,p) EDFT (T 0) (T ,p0 ) kT ln(p

p0)

G solid(T ,p) EDFT (T 0) Fvib(T ) PVX

Fvib : Stretching and bending

modes of the C-O groupsF

vib(T

)

(T)

C=O

C-O-C

O=C-O-C=O

O-C=O-O

C-O-C=O


• The defect free basal plane is inert towards O2.

Oxygen adsorption of graphene

O2 adsorption

O diffusion


• The pentagons were identified as the active sites for O2-dissociation.

Oxidation of carbon materials

O2 adsorption

O diffusion

Equilibrium vacancy structure during oxidation

V3:3O

V1:2O

V5:4OV2:2O

V4:3O

Vacancies are dominated by C-O-C and C=O groups at low pO2 . Atmospheric

pressure also forms O=C-O-C=O.

T=650 C

V6:4O


• Additional O2 adsorption forms larger groups, such as C-O-C=O (Lactones) and O=C-O-C=O (Anhydrides).

O2-adsorption at an O-saturated vacancy

V4:3O+2Otop V4:5O (Lactone) V4:7O (Anhydride)

O2 adsorption

O diffusion

V4:30


Two step mechanisms for oxidation:

1: Saturation of undercoordinated C-atoms by O2-dissociation

2: Additional O2-dissociation activates the CO2-desorption.

JMC et al, Phys. Rev. Lett 102, 166104 (2009)


Fundamental questions about graphene

• Can graphene exist as a true free-standing 2D-crystal?

• Do the electrons in graphene really behave as massless relativistic particles?

Relativistic space-time cone


Graphene: a perfect 2D crystal?

The Mermin-Wagner theorem says that 2D crystals are unstable at finite temperature!


• The TEM observations indicate that individual graphene layers are buckled at finite temperature!

• Real or experimental artifact?

Experimental observations of Graphene at finite T

TEM-picture of a single Graphene layer placed on a TEM-gitter,

Meyer et al., Nature 446, 60(2007)


Preparation:

• Enter a large graphene sheet into a supercell

Simulations of graphene at finite temperature

•Atomistic Molecular dynamics simulations

•Materials Studio Forcite

•Atomistic Monte carlo simulations


MD simulations of graphene

T=298 K

T=1000 K

Large fluctuations perpendicular to the graphene layer.


• No restoring forces perpendicular to the graphene sheet-> Soft modes.

Phonons in Graphene

Mounet et al., Phys. Rev. B 71, 205214 (2005).

LA

TA

LO

TO

ZOZA


• Finite temperature Monte Carlo Simulations of a single

graphene layer showed that the layer ist buckled at finite

temperatures.

• The buckling is due to that there are no restoring force

perpendicular to the graphene layer.

Simulations of graphene at finite temperture

Fasolino et al., Nature Materials 6, 858 (2007).

JMC, Nature Materials 6, 801 (2007).


Relativistic effects in a pencil trace?

Relativistic particles obey the Dirac equation:

such that the energy has the form

E2=m2c4+p2c2,with a gap mc2

but massless particles has linear dispersion without a

gap.

B

A

B

Amc

x

ii

ix

i


• The band structure of graphene at the Fermi level has the same dispersion as a mass less Dirac particle if c is exchanged for vF!

Comparison of the Dirac spectrum to graphene

Relativistic particles Electrons at the Fermi level in graphene


• Graphene has a number of unique properties as the first truly 2D material.

• There are many promising applications for graphene, but it remains to be seen which of these that will have market success.

• Simulations can give insight into important properties for device engineering.

Summary

unraveling the secrets of graphene by multiscale simulations

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