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PHYS824: Introduction to Nanophysics GNRs and CNTs

Graphene Nanoribbonsand

Carbon Nanotubes

Graphene Nanoribbonsand

Carbon Nanotubes

Branislav K. NikolićDepartment of Physics and Astronomy, University of Delaware,

Newark, DE 19716, U.S.A.

http://wiki.physics.udel.edu/phys824


From Bulk Graphene to GNR

11-AGNR: Empty circles denote hydrogen atoms passivating the edge carbon atoms, and the black and gray rectangles represent atomic sites belonging to different sublattices of the honeycomb lattice of graphene. The 1D unit cell (or supercell) distance and ribbon width are represented by da and Wa, respectively. The

carbon-carbon distance on the nth dimer line is denoted as an.

6-ZGNR:The empty circles and rectangles follow the same

convetion as for AGNR. The 1D unit cell (or supercell) distance and the ribbon width are denoted

as dz and Wz, respectively.


From Bulk Graphene to CNT

( , )

1 2

(1 ) 3 1ˆ ˆ

2 2

n ma n m na

κ κ⊥

+ −= + = +

a a x y

( , )

2

1 (1 ) 3 3ˆ ˆ

2 2 1

n ma a

κ κ λ

κ κ

− += +

+ + x y

�

smallest rational number that produces

as the graphene lattice vectorλ =

( , )n ma�

n mκ =

Terminology: (n,0) – zigzag CNT | (n,n) – armchair CNT | (n,m) n≠m chiral CNT


Experimental Methods to Fabricate GNRScience 319, 1229 (2008): Chemical Derivation

Nature Nanotechnology 3, 397 (2008): STM Nanolithography Nature 458, 872 (2009): SWCNT Unzipping


How to Compute Subband Structure of Quantum Wire Defined on a Lattice

ii--11 ii i+1i+1


Application to GNR: Supercells and Block Matrix Structure of TB Hamiltonian


GNR in Equilibrium: Subband Structure and Density of States

ARMCHAIR GNR

ZIG

ZAG GNR

4a

N = 5a

N = 30a

N =

4z

N = 5z

N = 30z

N =


20 15 10 5 0-3

-2

-1

0

1

2

3

Nz=20

Ener

gy

Conductance (2e2/h)

∆∆∆∆E=0.42

zigzag

-1 0 1-3

-2

-1

0

1

2

3

kx (1/a)

10 5 0-3

-2

-1

0

1

2

3

Na=21

Ener

gy

Conductance (2e2/h)

∆∆∆∆E=0.14

armchair

-1 0 1-3

-2

-1

0

1

2

3

kx (1/d

a)

GNR out of Equilibrium: Conductance Quantization Theory

1

2 1n

z

E t nN

π = ± +

−


GNR out of Equilibrium: Conductance Quantization Experiment

van Weeset al., PRL 60, 848 (1988)

Avouris Lab, PRB 78, 161409(R) (2008)


Pseudo-Diffusive Transport in Gated ZGNR

Crestiet al., PRB 76, 205433 (2007)


Electronic Structure of CNT: From Graphene via BZ Folding Method

Metallic 1D energy bands are generally unstable under a Peierls distortion → CNT are exception since their tubular structure impedes this effects making their metallic properties at the level of a single molecule

rather unique!

2n+m ≠ 3ln≠3psemiconducting

2n+m=3ln=3pall metalicmetallic

chiral(n,m)

zigzag (n,0)

armchair (n,n)

True band structure, as obtained from DFT, is also

affected by the curvature of the CNT and variations in the bond lengths which are not all equivalent. The effect are more pronounced in CNT of small

diameter, but they do not alter significantly the simple picture based on electronic structure of

graphene.


CNT out of Equilibrium: SubbandStructure and Density of States

Charlier, Blase, and Roche, Rev. M

od. Phys. 79, 677 (2007)

(5,5) (9,0)

(10,0) (8,2)

⇒


CNT out of Equilibrium: Conductance Quantization Theory

Charlier, Blase, and Roche, Rev. M

od. Phys. 79, 677 (2007)

Infinite metallic (5,5) armchair CNT

CNT-CNT Junction


CNT out of Equilibrium: Conductance Quantization Experiment


GNR and CNT Subband Structure: Tight-Binding vs. DFT


ZGNR Conductance: Tight-Binding vs. DFT


Band Gaps in AGNRs:Tight-Binding vs. DFT


Band Gaps in ZGNRs:Coulomb Interaction Effects

1 2 3 4 5 6 7 8 9 100.0

0.4

0.8

1.2

1.6

-3 -2 -1 0 1 2 30.0

0.4

0.8

1.2

1.6

Loca

l D

OS

at

EF=

0.0

1

Transverse Lattice Site

DO

S

Fermi Energy EF

( ) 1F

Ug E >

Stoner criterion for itinerant ferromagnetism


Magnetism in Graphene Clusters with Zigzag Edges: LSDA or Hubbard Model

Pisani et al, PRB 75, 064418 (2007)


Magnetic NanographenesHave Midgap (E=0) States

1 E=0 state

S=1/2

Courtesy of J. Fernández-Rossier, Universidad de Alicante, Spain

S=12 E=0

states


Midgap (E=0) States vsSublattice Imbalance


1 E=0 state

S=1/2 N=19,

NA=9

NB=10

2 E=0

statesS=1

N=22,

NA=10

NB=12

Number of E=0 states in nearest-neighbour TB model on bipartite lattice: Nz=|NA-NB|

M. Inui, S. A. Trugman, and E. Abrahams, PRB49, 3190 (1994).

Zero energy states are sublattice polarized(in majority sublattice).

Global sublattice imbalance: |NA-NB|>0


Properties of E=0 States on Triangles


JFR , J. J. Palacios, PRL 99, 177204 (2007)

100% Sublattice polarized

Predominantlyon the Edge

N=97,

NA=45

NB=52

7 zero energy states:

7 E=0 state


Electronic Structure in Non-Interacting Picture


7 electrons in

7 states


Electronic Structure in Interacting Picture


Superatomic Hunds’s rule

S=7/2


No Sublattice Imbalance S=0 Systems


0% Sublattice polarized

Low energy states Predominantlyon the Edge

No strict zero energy states

Low energy edge states

JFR , J. J. Palacios, PRL 99, 177204 (2007)

Large Hexagons have ferro-magnetic edgesTotal spin S=0 (AF inter-edge coupling)Small hexagons have no local moments

Competition between Coulomb interaction and AB hybdrization


GNRs as Topological Insulators Below 0.5 K

How big is SO coupling in graphene? Answer: arXiv:0904.3315

GNR with

intrinsic SO coupling

→2D Topological or Q

uantum Spin H

all Insulator


Anticipated Applications of GNRs and CNTs: Nanoelectronics with GNRFETs or CNTFETs


Nonequilibrium Phase Transition in ZGNRs as Potential Digital Memory

Dai Lab, Science 319, 1229 (2008)

Areshkin and Nikolic, Phys. Rev. B 79, 205430 (2009)

NEGF-MFA Hubbard

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