Transcript Carbon nanotubes - Northwestern University Mesoscopic

Carbon nanotubes
Mesoscopic group meeting Sept 17, 2003
One dimensional channels
Conductance of a one dimensional channel
Number of channels
Conductance of a ballistic conductor
Spin
(Actually, resistance of 1D channel is 0 in a 4-terminal measurement,
finite conductance arises from impedance mismatch to 2D probes)
For single wall carbon nanotubes, n is 2, so conductance is 4e2/h
With scattering, finite transmissions Tn
Single Particle States: The Coulomb blockade of charge
Consider a capacitor C, ask how much energy it takes to
charge it with a single electron charge e
e2
Ec 
2C
Energy is charging energy
e
C
For typical capacitors (1 nF), Ec is small (10-10 eV!)

What about nanometer
scale capacitors?
For practical reasons, it is easier to make devices that look like
Source
Drain
nanoparticle
Model as two ‘leaky’ capacitors in series, so that electrons can
tunnel into and out of central island from either electrode
The Coulomb blockade of charge
e
Energy required for 1 electron to tunnel onto central
island (nanoparticle) is
e2
Ec 
2C
What is the capacitance of the nanoparticle?

Estimate from self-capacitance C=4pe0r (r = radius of particle)
With r~10 nm, C~10-18 F (1 attoFarad)
Each quantum electron energy level is separated by Ec~80 meV>> kBT
Ec
EF
e
Source
Drain
Single Electron Transistors
No conduction unless nanoparticle levels line up with Fermi energy of source
Ec
and drain electrodes
Ec
Ec
e
E
EF
e
Source
Vg
Add third gate electrode to tune energy levels
Conduction is high when nanoparticle levels
line up with source and drain Fermi energy;
low otherwise
Transistor action, very sensitive 3-terminal
device
Most sensitive electrometer
Typical charge noise 10-4 e/Hz1/2
c
Ec
Ec
Ec
Ec Drain
Resonant tunneling
Quantum dot attached to two leads.
Typical tunneling is sequential incoherent tunneling
For coherent tunneling, form a resonance between localized
electron state and free electron states in source and drain
(Breit-Wigner resonance)
Ec
EF
e
Source
Drain
Conductance is symmetric about localized state energy e0
G is the intrinsic linewidth, temperature broadening linear in T
Conductance through 1D channels: Luttinger Liquids
For 1D channels with strong interactions, Fermi liquid theory
breaks down
Leads to power law correlations in representative quantities
Interactions characterized by an interaction parameter g
g is just the dimensionless conductance (in units of e2/h).
For g<1, interactions are repulsive, for g>1, interactions are attractive
g=1 corresponds to no interactions, or the Fermi liquid state
For 2D systems, we still have a Fermi liquid
For a 1D wire of length L attached to 2D contacts, LL behavior is
cut off at length scales longer than L. Corresponds to an energy scale
True 1D channels?: Magnetic edge states in 2DEGs
Milliken, Umbach and Webb, SSC 1996
Known value of interaction parameter g
n1 state, conductance is e2/h (g=1)
n1/3 state, conductance is e2/3h (g=1/3)
Luttinger liquids in carbon nanotubes
Bockrath et al, Nature, 1999
Fano resonances
Interference between direct transmission and transmission via a
localized state T=|t|2
Transmission through resonant
state
Direct transmission
Total transmission is
Write this in the usual Fano form
with