Michael S. Fuhrer University of Maryland

Graphene: Scratching the Surface

Michael S. FuhrerMichael S. FuhrerProfessor, Department of Physics andProfessor, Department of Physics and

Director, Center for Nanophysics and Advanced MaterialsDirector, Center for Nanophysics and Advanced MaterialsUniversity of MarylandUniversity of Maryland

Michael S. Fuhrer University of Maryland

Carbon and GrapheneCarbon and Graphene

C-

--

-

Carbon Graphene

4 valence electrons

1 pz orbital

3 sp2 orbitals

Hexagonal lattice;1 pz orbital at each site

Michael S. Fuhrer University of Maryland

Graphene Unit CellGraphene Unit Cell

Two identical atoms in unit cell: A B

Two representations of unit cell:

1/3 each of 6 atoms = 2 atoms

Two atoms2a

1a

Michael S. Fuhrer University of Maryland

Band Structure of GrapheneBand Structure of Graphene

Tight-binding model: P. R. Wallace, (1947)(nearest neighbor overlap = γ0)

2cos4

2cos

23cos41)( 2

0

akakakEE yyxF k

kx

ky

E

Michael S. Fuhrer University of Maryland

Bonding vs. Anti-bondingBonding vs. Anti-bonding

00

0 0

0

EH

ψ “anti-bonding” anti-symmetric wavefunction

“bonding” symmetric wavefunction

022 11

21

E

011 1

12

1

E

γ0 is energy gained per pi-bond

Michael S. Fuhrer University of Maryland

Bloch states:

AB

AB

01

10

FA(r), or

FB(r), or

“anti-bonding”E = +3γ0

“bonding”E = -3γ0

11

21

11

21

Γ point:k = 0

Band Structure of Graphene – Band Structure of Graphene – ΓΓ point ( point (kk = 0) = 0)

Michael S. Fuhrer University of Maryland

34

32

1

i

i

e

e

λλ

λ

K

K

K

01FA(r), or

10FB(r), or

Phase:

K 23a

a3

4K

Band Structure of Graphene – K pointBand Structure of Graphene – K point

Michael S. Fuhrer University of Maryland

34

32

0 1

i

i

i

e

e

e

Phase:

Bonding is Frustrated at K pointBonding is Frustrated at K point

32

02

ieE

001ieE

34

03

ieE

0

034

32

00

iii eeeE

32

02

ieE

001ieE

34

03

ieE

0034

32

00

iii

i eeeeE

Re

Im

E1

E2

E3

Michael S. Fuhrer University of Maryland

01

FA(r), or

10

FB(r), or

K

23a

a3

4K

0π/3

2π/3π

5π/3

4π/3

“anti-bonding”

E = 0!

“bonding”

E = 0!

11

21

11

21

K point:Bonding and anti-bonding

are degenerate!

Bonding is Frustrated at K pointBonding is Frustrated at K point

Michael S. Fuhrer University of Maryland

)()()( rrvF FFkσ

kvbeibe

ek Fi

ii

k

k

;2

12/

2/rk

θk is angle k makes with y-axisb = 1 for electrons, -1 for holes

Eigenvectors: Energy:

Hamiltonian:

)()(

)()(

00

rFrF

rFrF

ikkikk

vB

A

B

A

yx

yxF

electron has “pseudospin”points parallel (anti-parallel) to momentum

K’

K

linear dispersion relation“massless” electrons

Band Structure of Graphene: k·p approximationBand Structure of Graphene: k·p approximation

Michael S. Fuhrer University of Maryland

Visualizing the PseudospinVisualizing the Pseudospin0

π/3

2π/3π

5π/3

4π/3

180 degrees

540 degrees

Michael S. Fuhrer University of Maryland

Visualizing the PseudospinVisualizing the Pseudospin0

π/3

2π/3π

5π/3

4π/3

0 degrees

180 degrees

Michael S. Fuhrer University of Maryland

K’ K

K: k||-x K: k||xK’: k||-x

real-spacewavefunctions(color denotesphase)

k-spacerepresentation

bondingorbitals

bondingorbitals

anti-bondingorbitals

Pseudospin: Absence of BackscatteringPseudospin: Absence of Backscattering

bonding

anti-bonding

Michael S. Fuhrer University of Maryland

““Pseudospin”: Berry’s Phase in IQHEPseudospin”: Berry’s Phase in IQHE

π Berry’s phase for electron orbits results in ½-integer quantized Hall effect

-80 -60 -40 -20 0 20 40 60 800

5

10

15

20

-34-30-26-22-18-14-10-6-22610141822263034

xy (e

2/h)

QHE at T=2.3K, B=7.94T

Rxx

(k)

Vg (V)

214

2

nhe

xy

422 vsgg Berry’s phase = π

holes

electr

ons