Ch120a Fall 2008 Lecture - California Institute of...

MATERIALS AND PROCESS SIMULATIONCENTER,CALIFORNIA INSTITUTE OFTECHNOLOGY, PASADENA, CA 91125

Ch120a Fall 2008 Lecture

TODA. PASCAL MSC, CALTECH

Recap

Ground state of Carbon

Ethane

DiamondDiamond

C C multi bondsC –C multi‐bonds

Ethylene

Benzene

Can form periodic systems with carbons with 1200 bond angles


1Πu


1Πu

C3 C4

C5 C6Circum‐coronene


In Short Graphene is a sheet of carbon (monolayer of graphite), just one

atom thick Atomically thin zero‐gap semiconductor with linear dispersion Atomically thin zero gap semiconductor with linear dispersion

of conduction and valence band electrons Already used to test predictions of relativistic quantum

mechanics, graphene is showing promise in a diverse range of applications such as sensors, transistors and batteries

Chemically modifying graphene could tailor its electrical y y g g pproperties, or help it to form strong composites

Curling, rolling and stacking can turn grapheneinto buckyballs, nanotubes ‐ and graphitey , g p


Wallace Phys Rev 71 622 (1947)Wallace, Phys. Rev. 71, 622 (1947)

Graphene‐based field‐effect transistor: GraFET

Geim and Novoselov, Nature Mat. 6, 183 (2007)


30

40

/mol)

0

10

20

6ng ene

rgy (kca

l/

be

XYG_be

Be_cp

avg be

‐30

‐20

‐10 2.5 3.5 4.5 5.5 6.5 7.5

Bindin

Plane‐Plane separation (Å)

avg_be

Counterpoise calculation with Jaguar v7: M‐06 2X/6‐311G**+

a e a e sepa at o ( )

M 06 2X/6 311G + Ultra fine grids and SCF accuracy = accurate BE = Dimer – (A + B) BE(cp) = BE – cp corrections

▪ (A_cp –A_no_cp) + (B_cp – B_no_cp)Di l ll l h t Å Å Displace parallel coronene sheets: 2.73Å ‐> 7.03Å Calculate singlepoint of dimer Calculate A_cp/B_cp to obtain CP corrections Average B.E = 0.5 (BE + BE(cp))


‐3‐2.2‐1.4‐0.6

0.21

1.82.6

‐12

‐3

‐2.7

‐2.4

‐2.1

‐1.8 1.5

1.2

0.9

0.6 .3 0 3 6 9 2 3

‐14

‐13

‐ ‐ ‐ ‐ ‐ 1 ‐0 ‐0 ‐0 0. 0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7 3

‐13‐‐12

‐15

‐14‐‐13

‐15‐‐14

‐16‐‐15

‐17‐‐16

‐17

‐16 ‐18‐‐17

6 minima found on PES

‐18

Y = 0, x = ± 1.3; y = ± 1.7, x = ± 2.3 Minima = 19.3 kcal/mol Barrier: 6.3 kcal/mol


“Discovered” by Iijima(1991)/Smalley (1993)

Can be metallic/semi‐ Can be metallic/semi‐conducting

Arm chair/zig‐zag/chiral Single walled (SWNT) or

multi‐walled (MWNT) Synthesized by several

methods Arc discharge Laser furnace

Ch i l

STM images of carbon nanotubesT.W. Odom, J.‐L. Huang, P.Kim, C.Lieber, N t ( 8) Chemical vapor

depositionNature 391 (1998)


The structure of SWNT can be represented based on 2D graphene lattice.

OA and OB define the chiral OA and OB define the chiral vector Ch and the translational vector T of the nanotube.

The rectangle OAB'B defines the unit cell for the

Ch = na1 + ma2 = (n,m)

nanotubes.

The chiral angle θ defined as

with 0<θ<300

The diameter of the carbon nanotube, dt


SWNT exhibits either metallic or semiconductor properties depending only on SWNT exhibits either metallic or semiconductor properties depending only on tube diameter and chiral angle.

The metallic conduction occurs whenn – m = 3q

All armchair nanotubes are expected to be metallic, while one‐third of zigzag and p , g gchiral nanotube should be metallic, with the remainder being semiconducting.


For SWNT, cyclic or periodic boundary condition in the rolling direction only t t th d li i th states on the red lines in the 2D BZ can exist. (k=2Nπ/R)

If K points are on the lines mod(n‐m)≡n‐m‐(interger)=0 no energy gap

lli b metallic nanotubes If K points are not on the

lines mod(n‐m)=1 or 2 energy gap appears

i d ti semiconducting nanotubes


Diameter dependence: Smaller diameter, larger Eii, g ii

n‐m (chiral angle) dependence: Semiconducting:▪ Eiimod(n‐m,3)=1 < Eiimod(n‐m,3)=2 , i = odd integer.g

▪ Eiimod(n‐m,3)=1 > Eiimod(n‐m,3)=2 , i = even integer.

▪ Differences increase with n‐m value.Differences increase with n m value.

Metallic:▪ Degeneracy with n‐m=0. Peak splitting increases with n msplitting increases with n‐m.

Ch120a Fall 2008 Lecture - California Institute of...

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