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3. NANOMEMS PHYSICS: Quantum Wave Phenomena 91

sub-bands or “channels” in which transport can only occur once the electrons
acquire the corresponding necessary energy, in other words, electrons behave
as waves with discrete (quantized) wave vectors. The quantized electrical
conductance is a manifestation of this. In contrast, electrons in TLs of
relatively large dimensions may exist at virtually all energies and, if there
were no interaction among electrons, they would behave as free particles.
The theory of electron behavior in a metal, when electron-electron
interactions are taken into account, is due to Landau [131] and is denoted
Fermi liquid theory. A Fermi liquid is considered to be made up of “quasi-
particles,” which are fictitious entities that, while being physically different
from electrons, behave similarly to electrons, but with a different mass and
dispersion relationship.
When electron transport is confined along one dimension, a behavior
different to that of free electrons and that of a Fermi liquid is observed. The
new aggregate of entities is said to consist of another fictitious quasi-
particle, namely, the plasmon, and is referred to as a Lüttinger liquid (LL).
The distinction between Fermi liquid and Lüttinger liquid behaviors is
important to the realization of nanoscale circuits and systems, not only from
the point of view of TL properties, but also because their different behavior
elicits new issues when connecting a Fermi liquid TL to a Lüttinger liquid
TL. The fundamental aspects of Fermi and Lüttinger liquids are addressed
next.

3.1.4.2.1 Fermi Gas

The Fermi liquid theory explains the success of the free-electron
approximation in the calculation of transport problems, even in the context
of electron-electron interactions. The usual point of departure for describing
the Fermi liquid is the Fermi gas. This is the conceptual situation in which
the metal is modeled as a solid of volume V and length L on a side
( V = L 3 ), which contains moving non-interacting electrons in much the

same way as atoms and molecules move inside a gas container. Since the
electrons are assumed to be independent, i.e., do not interact, they each obey
a Schrödinger equation of the form [132],

ª p 2 G º
H ψ = + U () ψr = E ψ , (30)
0 « »
¬ 2 m ¼
G
where the potential energy is taken to be () 0=rU . The solution of this
equation is then obtained by assuming that all space is filled by cubes of side