ABSTRACT
The electronic properties of matter are of fundamental relevance
for the function and behavior of the physical world as we know
it. Normal matter is built up of atoms, where literally all mass is
focused in the nucleus, but the majority of its interaction with the
local environment is mediated by the engulfing cloud of electrons.
The mutual electric interaction between the individual atoms, or
more precisely between bound electrons, is the key ingredient
for accessing the intrinsic physical properties of matter. On a
very fundamental level there are throughout physics two different
descriptions of bound electrons. Very interestingly, we have to
combine, but not mix, these two opposing concepts for the complete
description of electrons in carbon nanotubes. The first case is bound
and localized electrons, as for instance in an isolated atom. There the
electron can exist only in discrete states with well-defined quantum
numbers. The discrete electronic transitions between this states
give rise to the emission and absorption spectra of glowing gases.
The second case is bound but delocalized electronic states. These
exist within condensed matter, where the electrons can behave
as quasi-free particles. Here they are allowed to propagate and
possess momenta that correspond to continuous energies [Kittel
(1963)]. Still, the connection between energy and momentum is no
longer a parabola as in homogeneous free space, but it gets strongly
modified by the discrete crystal structure inside matter. An electron
in free space has a constant rest mass, which just adds to its kinetic
energy, but in a solid there are additional energetic contributions
stemming from the interaction of the electron with the lattice.
The actual momentum of an electron determines the wavelength
of the corresponding electronic wavefunction and thus also the
spatial overlap of the electron with the surrounding crystal. The
electronic dispersion relation, viz. the electrons energy as a function
of their momentum, is called the electronic band structure. Typically
it consists of several branches that originate from the different
symmetries of the allowed electronic wavefunctions. The material
specific shape of the band structure determines the electronic
density of states (DOS). The DOS just tells how many electronic
states can be there per unit volume with a certain energy, regardless
of their actual momentum or their spin state. In an isolated atom
the DOS is a discrete set of infinitesimally sharp peaks (δ functions),
but in solids the DOS, which is readily derived by taking the
inverse slopes of the dispersion relation, is a smooth function. In
a two-dimensional sheet or in a one-dimensional wire the DOS
is a staircase function or a series of sharp van Hove singularities
(VHS). The latter one-dimensional VHS are a fingerprint of truly one-
dimensional electronic systems. The general shape of the energy-
dependent DOS in one, two, and three dimensions is recapitulated
in Fig. 4.1.
