Ballistic and quantum transports in carbon nanotubes

Discrete energy levels in carbon nanotubes

Two atoms two energy levels

Three atoms three energy levels

atoms

CB

VB

Metallic(no band gap) Semiconductor

Small band gap (0.1-3eV) Insulator (large band gap > 5 eV)

N = infinite (metals)

EF

Ve-

Free-e-

VB

CB

Spacing between levels becomes too small to be distinguishedSo it can be regarded as a band structure

Fermi sea

Free-e-

As long as kinetic energy is sufficient free electron movementcan change from lane to lane

a. Underlying mechanism for ballistic transport

Bulk Cu

Corresponding band structure

CB

VB

Conduction electron paths in all directions within CB

EF

Nanowire (quantum wire)

Nanodot (quantum dot)

EF

EF

Discrete levels

Sub-bands

Spacing increase

Band gap

M

EF

temp

Science, 283, 52 (1999).

Quantum transport in carbon nanotubes

a. Metallic CNTs have two conduction bands (two conduction channels). b. one conduction channel of quantum unit = Go = 12.9 (K)-1 = 2e2/h.c. Two conduction channels = 2Go = 4e2/h = 6.5 (K)-1

If contact resistance is small and negligiblethen CNT resistance between contacts Should be 6.5K

In practice, resistance exceeds 6.5 k and underlyingMechanism comes from

A. Semiconducting tubesB. Huge contact resistanceC. Defects in metallic CNTs

Electrons are unable to enter into CNT (joule heating)

High contact resistance

Thick contact barrier

Thin contact barrier

CNT

CNT

+-Electric charges induce voltage at leads but no current flow in CNT (coulomb oscillation)

tunneling

Existing conduction electrons in CNT

Existing electrons in CNTs exclude incoming electrons

Coulomb blockage

Thin contact barrier

CNT

tunneling

Gate voltage

Gate voltage

Gate voltage: electric field modulatedchemical potential (energy levels)

gate

Two electrons occupying one level

gate

Modulation of energy levels by gate voltage

Go

Ballistic transport in carbon nanotubes

Electron conduction has no resistanceand no heat generation and structures are defect-free.

e-

Conductor (e.g. Cu)

e-

Transport delay by resistance

e-

Resistance point (scattering)

Resistance comes from thermal vibration of crystal lattice, electrons and impurities

forward scatteringbackscattering

mean free pathrelaxation time

mean free path: no resistance

Cu (mean free path) = 1 m

Size reduction to below 1 m in length

scattering pointscattering point

Cu nanowiree-

free path with no resistance

Nanowire on electrodes

electrodes

resistance at wire-electrode contact

e-

forward scatteringforward scattering

backscattering

Fabry-Perot interference

When defects exist the ballistic transport is absent

Fabry-perot interference

defect

No Fabry-perot interference

Defects in CNTs

EF

Scattering center

Blocking of two conduction channels此時碳管電阻值昇高

Mean free path in Cu is 1 m

1m

Mean free path is ca. 100-300 nm in CNT

Scattering centers

Conditions for ballistic transport

a. Tube length electron mean free paths (or no defects)

b. Low contact resistance (low capacitance)

+ -

High contact resistance (capacitor-like)

CNT

capacitor

leads

dielectric

c. Gate voltage is not needed (different from that of quantum wire)

gate

High contact resistance (tunneling) Low contact resistance

Science, 280, 1744

R

+

-

R

Distance

A

A

R

+

-

B

R

+

-

C

B C

A general case

Ballistic transport effects

a. No heat generation, because no electron-phonon interaction (i.e. no scattering by defects)

b. Stepwise I-V profile (or quantum conduction)

Conduction via individual atoms

a. Nano-contact

b. electro-sharpening of metal wire

Diffusive conduction Quantum conduction

c. Mechanical break junction

Single atom電極

電子

How to transmit through a single atom

Conduction through individual orbits

3

6

5

Ohmic conductor (linear I-V profile)

metals

Voltage

Current

Non-linear I-V profile (non-ohmic conductor)

voltage

current

Light bulb

semiconductors

voltage

current

Theory of coulomb blockage

source drain

nanotube

If one transfers the charge Q from the source to the grain the change in the energy of the system is

                                                         

http://edu.ioffe.ru/register/?doc=galperin/l13pdf1.tex

the first item is the work by the source of the gate voltage while the second is the energy of Coulomb repulsion at the grain.

the effective capacitance C

the gate voltage VG

source drain

nanotubee-

-+Polarization of leads

Q  = –CVG    So Q can be tuned by the gate voltage VG

the charge is transferred by the electrons with the charge –e.

Then, the energy as a function of the number n of electrons at the drain is

the difference

at certain values of VG,

and the difference vanishes.

It means that only at that values of the gate voltage resonant transfer is possible.