Page 136 - Science at the nanoscale

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Low-Dimensional Nanostructures
126
ballistically is given by:
2
2e
(6.19)
G =
h
where e is the electron charge and h is Planck’s constant. Hence,
1D ballistic electron transport is quantised and the quantum of
2
resistance R = 1/G = h/2e ≈ 12.9kΩ. In the absence of colli-
sions, the resistance can only originate from the conductor-contact
interface, and hence R is often called the contact resistance. Ohm’s
law implies that the conductance is inversely proportional to the
length of the sample, but the conductance of ballistic structures is
independent of the length of the sample. Such 1D quantised con-
ductance has been observed in individual multiwall carbon nan-
4
otubes (Fig. 6.10).
Electron transport at the nanoscale depends on the relationship
between the sample dimensions and three important characteris-
tic lengths:
1. The mean free path L , which represents the average
f p
distance an electron travels before it collides inelastically
with impurities or phonons;
2. The phase relaxation length L , which is the distance
ph
after which the phase memory of electrons, or electron
coherence, is lost due to time-reversal breaking processes
such as dynamic scattering;
3. The electron Fermi wavelength λ F , which is the wave-
length of electrons that dominate electrical transport.
The electron transport is diffusive if L > L . The transport is
f p
ballistic if the sample length L ≪ L , L , i.e. the electron does ch06
f p
ph
not scatter and the electron wave function is coherent. In mod-
ern high-mobility semiconductor heterostructures, L f p and L ph
can be tens of micrometers; on the other hand for polycrystalline
metal films L f p is just tens of nanometres. The conductance G is
2
quantized G ∼ e /h when L ∼ λ F . Diffusive transport involves
electrons with a wide energy distribution, but ballistic transport
involves only electrons close to the Fermi energy, E F .
The quantisation of an electron’s resistance can be understood
in semi-classical terms. When a voltage V is applied between
4 S. Frank et al. Carbon nanotube quantum resistors, Science 280, 1744–46 (1998).