Graphene and Carbon Nanotubes

One atom thick layer of graphite = ‘graphene’

1 atom thick films of graphite – ‘atomic chicken wire’

Novoselov et al -Science 306, 666 (2004)

Geim’s group at Manchester

Novoselov et al - Nature 438, 197 (2005)

Kim-Stormer group at Columbia University

Zhang et al - PRL 94, 176803 (2005)

Zhang et al - Nature 438, 201 (2005)

100μm

Scanning Transmission Electron micrograph of GRAPHENET

hank

s to

Dr

Nic

olos

i (O

xfor

d M

ater

ials

)

Tight binding Calculation: Graphene

antibonding

antibonding

Tight binding Calculation: Graphene

NB, Two C atoms per unit cell (ie 12 electrons, 8 of which are valence)Non-hybridized (pi) electrons play key role in graphene conductivity

Graphene bandstructure

2N

2N

2N

2N

Tight Binding Theory

* 0 032K

ac

0

* * 1

1K E

c c

cK* = 1 x 106 ms-1

Typical values of are in the region of 3 eV, β = ~0.1, giving:

/ 1.005c c 3 million k.p.h.

Minimum conductance at K

(Dirac)-point

• Conductance minimum as

Fermi energy passes through

zero density of states

• Manchester and Columbia

groups

h

e

lkh

e

mv

lnene F

F

2

22

Mott criterion: kFl ≈ 1

X 4

A graphene based future

Graphene predicted to lead to lots of new physics +New fast transistors, super strength materials, transparent electrodes, chemical sensors.....

Single walled Carbon nanotubes:Discovered in 1993!

(10,10) Armchair Carbon Nanotubes

Carbon nanotubes:rolled up graphene!

http://www.photon.t.u-tokyo.ac.jp/~maruyama/agallery/agallery.html

(0,0)

Ch = (10,5)

Wrapping (10,5) SWNT

a1a2

x

y

Carbon Nanotubes (CNTs)Chiral vector for

Tube diameter

Chiral angle (definition)

Number of hexagons in nanotube unit cell:

Greatest common devisor of

CNT “translation” vector

CNT

1st lattice point reached!

(5,2)

(n,0) are called “zigzag” nanotubes(n,n) are called “armchair” nanotubes

Chiral vectors are used to label CNTs:

Carbon Nanotubes

Cyclic boundary conditions give allowed k-states for CNT:

Where reciprocal lattice vectors for CNT unit cell:

(circumference recip. lattice vector)

(CNT axis recip. lattice vector)

If allowed k-state coincides with graphene 1st BZ K point = metallic!

Number of CNT translationvectors along full length of tube

Allowed k-states for a (3,1) CNT

Extended Brillouin Zone Scheme Zone Folded Scheme

Tight binding Calculation: Graphene

antibonding

Tight binding Calculation: Graphene

antibonding

(7,4) (7,6)

metallic semiconducting

Carbon Nanotubes (examples)

Carbon Nanotube Bandgaps

Armchair CNT Zigzag CNT

SEMICONDUCTOR

Carbon Nanotubes (examples)

Zigzag CNT

METALMETAL

1,0

2,1

3,2

4,3

5,4

6,5

7,6

zigzag

armchair

2,0

3,1

4,2

5,3

6,4

7,5

8,6

8,7

4,0

5,1

6,2

7,3

8,4

9,5

5,0

6,1

7,2

8,3

9,4

7,0

8,1

9,2

10,3

8,0

9,1

10,2

10,0

11,1

11,0

Metallic

Semiconducting

Carbon nanotubes: metallic if n1 –n2 is a multiple of 3

NB 1-D CNT Brillouin Zone

CNT (1-D) Bandstructure

(8,8) metallic (8,0) semiconductingCNT (1-D) Density of States

Photo-Luminescence Excitation Mapping

Scan E22 excitation energy and measure emission from E11. Luminescence (fluorescence)

Low Dimensional Structures and Materials

• Artificial layered structures -

Quantum Wells and

Superlattices

• Electric or Magnetic Fieldsapplied in one direction.

Layers may be only a few atoms thick

HeterojunctionsEnergy levels for 2 different

semiconductorsEnergy line up at junction of

two (undoped) materials

Reduced Dimensionality

Quantum Well removes 1 Dimension by quantization

Electron is bound in well and can only move in plane

2-D system - motion in x, y plane

Quantum Mechanical Engineering

2 quantum wells give 2 levels (symmetric and antisymmetric combinations)

Superlattice generates a (mini)band

E

k/L0

Quantum Well - Type I

Typical Materials: 1: GaAs

(Eg = 1.5 eV)

2: (Al0.35Ga0.65)As

(Eg = 2.0 eV)

Energy levels are quantized in

z-direction with values En for

both electrons and holes

E = En + 2k2/2m*

1-D 2-D

Infinite well - Particle in a box

• 1-D Motion in z-direction

n = 1

n = 2

n = 3

L

E

dz

d

m

2

22

*2

System is Two-Dimensional when:

E2 - E1 > kT

162 meV 25 meV at 300 K

L

znAn

sin

*2*2

2222

m

k

L

n

mEn

Typical values L = 10 nm, me* = 0.07 me En = 54 n2 meV

Finite Well

even parity odd parity

n(z) Acos kz Asin kz |z| < L/2

Bexp[-(z - L/2)] Bexp[-(z - L/2 z > L/2

Bexp[+(z + L/2)] -Bexp[+(z + L/2)] z < L/2

• where:

• assume m1 = m2 k2 + 2 = k02 = 2m*V0/2

n

k

mV

m

2 2

10

2 2

22 2* *

boundary conditions:

wavefunction 1 = 2

probability current

A cos (kL/2) = B A sin (kL/2) = B

kA sin (kL/2) = B kA cos (kL/2) = -B

k2 tan2 (kL/2) = 2 k2 cot2 (kL/2) = 2

k2 sec2 (kL/2) = k02 k2 cosec2 (kL/2) = k0

2

cos (kL/2) = k/k0 sin (kL/2) = k/k0

zmzm

2

2

1

1

11

Graphical solution of finite Quantum Well

Well depth determines value of slope k0-1

Optical Properties

Absorption coefficient is proportional to the density of states:

~ 1/2

Modified close to the band gap due to ‘excitons’

3-D

2-D - Big Changes

Multiple Band gaps -Band gap shift -Sharper edge

For wide wells the sum of many 2-D absorptions becomes equivalent to the 3-D absorption shape (1/2)

Correspondence principle.

GaAs/Al0.35Ga0.65AsQuantum Well

absorption

• Sharp peaks due toexcitons

• peaks doubled due toheavy and light holes

Semiconductor lasers

Quantum Well laser

Fibre Optic Communications,

CD players, laser pointers

Forward biased p-n junction

Molecular Beam Epitaxy (MBE)

• Ultra High Vacuum evaporation of molecular species of elements (Molecular Beam)

• Epitaxy - maintaining crystal structure of the ‘substrate’ - which is a single crystal

Metal Organic Vapour Phase Epitaxy (MOVPE)

• Chemical reaction of elements bonded in volatile organic compounds

• e.g. (CH3)3Ga + AsH3

GaAs + 3CH4

• Reaction takes place on a heated substrate and growth is also ‘epitaxial’

Heterojunctions and Modulation Doping

New idea for superlattices and heterojunctions: Separate the dopant impurities from the electrons

Gated structures

• Place metallic electrode on

surface to apply variable

electric field

• gives a variable potential

and surface charge density

• Basis of MOS transistors as

well as controllable 2-D

systems.

Negative electrode potential repels electrons underneath leaving only a narrow 1-D channel of conducting electrons

Quantized Conductance in 1-D -the Quantum Point Contact

For a short 1-D structure there is no scattering - Ballistic transport

Calculate the current carried by the electrons by adding up the contributions from all carriers travelling in one direction.

For flow in one direction only (k > 0) the density of states is:

i.e. half the usual 1-D density of states, but with a factor 2 to account for spin degeneracy.

21

2 2

22

1 21 2

L m

d

* /

/

The electrons have velocity v = (2/m*)1/2. Therefore the current in the positive direction is:

Apply a voltage V along the 1-D channel to give a difference in chemical potential. This causes a net current to flow:

h

edgeJ

2)(v

0

• Therefore the conductance is:

• When there are p 1-D

subbands occupied (e.g. if the

1-D wire is wider) each

contributes one unit of

conductance

Vh

ed

h

ed

h

eJJJ

eV

Tot

2

00.

222

h

e

V

J Tot2

. 2

h

epTotal

2

2

B.J. van Wees et al,

Phys. Rev. Lett. 60, 848 (1988)