Electron diffraction on carbon nanotubes Marko Viršek adviser: doc. dr. Maja Remškar

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Electron diffraction on carbon nanotubes

Marko Viršek

adviser: doc. dr. Maja Remškar

21.11.06 Electron difraction on carbon nanotubes

Outline

• Transmission electron microscope:• Electron diffraction on graphite• Geometry of carbon nanotubes• Kinematical diffraction theory for carbon nanotubes• Simulations and experimental diffraction patterns• Other helical structures


Transmission electron microscope (TEM)TEM components:

• HV source

• Vacuum sistem

• Electron gun

• EM lenses

• Apertures

• Specimen holder

• Viewing screen

Basic TEM modes:• Imaging

• Selected area diffraction


Interaction of electrons with the sample

• Specimen thickness < 100 nm

• Electron energies ~ 100 - 400 keV

(unscattered + elastically Bragg scattered e-)

diffraction pattern

scattering:

nonuniform distribution of electrons

spatial distribution angular distribution

image

(unscattered e-)

σelastic~(Z e / V θ)2

mean free path of e- ~ 10 nm

• Electrons interact stronger than x-rays


Electron lenses

1/u + 1/v = 1/fu....object planev....image planef.....focal plane

General properties:• Changable strength (f) of the lense• Collecting from small angles• Limiting the resolution of TEM• Using apertures for selecting electrons


Selected area diffraction

diffraction mode

imaging mode

object plane sample

focal plane

(obj. aperture)

image of the

diffraction

image plane (SAD aperture)

image of the sample

Objective lens :

Intermediate lense object plane:

at image plane of objective lenseat back focal plane of obj. lense

Selecting TEM mode:

remove

remove


Elastic scattering

kK

K0

X-rays electrons

KK0

C

• Bragg: nλ = 2dhkl sinθB

λ - electron wavelength ~2,5 pm at 200 keV

• von Laue: k = K - K0= ha*+kb*+lc*

and |k|=1/dhkl

• Many points in electron diffraction: small λ + relrods


Electron diffraction on graphite

)(*)(

))(2(/)(

kk

k

AAI

lzkyhxiExpfNAF iiii

ihkl

• Structure factor for primitive celll:

• Intensity of diffraction waves:

even is l and 1 3m 2k h if f |F|

odd is l and 1 3m 2k h if f 3 |F|

even is l and 3m 2k h if f 4 |F|

odd is l and 3m 2k h if 0 |F|

22

22

2 2

2

(hk.0) spots

structure of hexagonal graphite

TED pattern

(00.1) forbidden

(00.2) allowed

(hk.o) allowed


Geometry of single-shell carbon nanotubes

3

ηXX’

Chiral vector:XX’ = L a1+M a2 ; L > 0

Circumference of the tube:|XX’| = 2πR0 = a

Chiral angle:tg η = M / (2L + M)

LM M L 22

• armchair: (L, L ); η = ±30º

• zigzag: (L, 0); η = 0º

• chiral: (L, M); −30° < η < 30°

rolling up


Geometry of carbon nanotubes

d

a = d 3

3 helical ribbons



30°+ |η|

L helical ribbons

M > 0: right handed tube

M < 0: left handed tubetube: (L > 0, M)

(4,1) nanotube



L paralel

zigzag helices

L paralel

double helices

a

u

z

After rolling up:

u

Ф

u = ФR0



222

211

/)2(2/3

/2/)2(

CMLCdMz

CMCMLdz

∆z1,∆Φ1 ∆z2,∆Φ2 =

function (a, L, M)

u

z

a

∆z1

∆u1

∆u2

∆z2



)2/()(2/

)/(2))(/2(22

000

0

0

MLLMMLpP

jPpzzP

pjzz

R

jj

j

j

Positions of carbon atoms on a single helix:

zigzag pair from primitive helix by a screw displacement (Δz1, ΔФ1)

L-1 pairs of helices from the first pair by (j Δz2, j ΔФ2), where j = 0,…, L-1

rj = (ρj, zj, Φj) = Function (R0, z0, Φ0, a, L, M)

Arangement of the atoms in the complete single-shell nanotube:

∆u2

∆z2

a

∆z1

∆u1


Kinematical diffraction theory Scattering amplitude for identical atoms: For a single primitive helix:

rj = (ρj, zj, Φj)

2y

2x

2kkk

Фk = arctg (ky / kx)

zk,k

j

ifA jrkkk exp)()(

)p

m

P

n(2kx

)2

(niexp)Rk(J)zkiexp()k(f)p/2()(A

z

n,m0k0n0z1 k

discrete values of kz layer lines

m, n integers


Kinematical diffraction theory

• The amplitude A2 (k) for a pair of parallel helices:

generated by screw displacement (Δz1, ΔФ1)

• The amplitude ASS(k) of the complete single-shell carbon nanotube:

generated by L-1 screw displacements (Δz2, ΔФ2)

)2)(

3

2(exp1)()( 12 msiAA

kk

l

zlSS lTkkFdCA )/2()()3/4()( k

N

MLm

N

MLslmsikfx

mMsLiRkJTlziF

c

smkmMsLl

)2()2(,3/)2(2exp1)(

)2/)((exp)()/2(exp)(,

000

k

where T translational period in z direction 2 independent integers

2 independent integers


Simulation of diffractionfor single-shell tubes for e beam normal

to the tube axis

b c(10, 10) armchair (36, 0) zigzag tube (18, 1) chiral tube

(nearly zigzag)graphite

• Zero order line represents zero order Bessel function• The oscilations represent slit function from upper/bottom tube edge

• Spots are not circular as in 3D crystals• Spots are diffuse streaks elongated normal to the tube axis and fading away


b c

(10, 10) armchair (36, 0) zigzag tube

2η

graphite

Simulation of diffractionfor single-shell tubes for e beam normal

to the tube axis

(18, 1) chiral tube (nearly zigzag)

• Two hexagonal patterns rotaded by η from z-axis

• Hexagonal (hk.0) pattern, rotated by 30 ° from armchair to zigzag


Multi-shell nanotubes

j

jjjjjSSMS zMLkAkA ),...,...,,,()( 00,

• First observations in 1991 by Iijima on multi-shell nanotubes:

• Multi-shell tubes contain coaxial single-shells of different chiralities:

• The amplitude AMS(k) of the multi-shell carbon nanotube:

7 layer nanotube Electron diffraction Simulation

7 tubules:

(29, 0) (38, 0)(47, 0) (48, 13)(55, 16)(63, 17) (70, 20)

7 zigzag and achiral tubules with η = 12°

24°

S.

Iijim

a, N

atur

e, 3

54,

56,

1991

j

jjjjjjSSMS zRMLakAkA ),,,,,,()( 000,


Diffraction pattern

• A constant honeycomb lattice along the axis

Sharp diffraction spots along the axis

• Shrinking lattice parameter along the

tubule circumference

Smaller lattice parameter – larger scattering angle

Spots are elongated away from the axis


Tilting experiment

(00.2) spots remain anafected

axis of rotation all other spots move

away from z= 0 axis

A, B, C, D climb up

and finnaly coincide

lattice distance is shrinkened


Tilting simulation

(25, 10) chiral tube, η = 16°

tilt angle θ: from 0° to 30°

distances between layer linesincrease like 1/cosθ

θ

1

cosθ

coalescence of spots beggins

at chiral angle!


Helical structures: DNA

Franklin R. E. and Gosling R. G., Nature 171, 740, 1953

Watson J. D. and Crick F. H. C., Nature 171, 737, 1953

DNA structure from

x-ray diffraction pattern

and CCV theory


Helical structures: WS2

and MoS2 nanotubes

WS2 nanotube revealing the

main chirality of 6.5° and 13°

Achiral Au–WS2 nanotube Au-WS2 nanotube

M. Remskar, Z. Skraba, C. Ballif, M. Regula, R. Sanjinés, F. Lévy, Adv. Mater. 10, 246, 1998


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Electron diffraction on carbon nanotubes Marko Viršek adviser: doc. dr. Maja Remškar