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S. Taioli
graphene, however, the electronic properties of single CNTs depend on their
chirality. The nanotube chirality is defined by the specific and discrete chiral angles
at which graphene sheets are rolled. In particular, the chirality is identified by a
couple of numbers (n, m), which define a vector (C h = na 1 + ma 2 ) in an infinite
graphene sheet that describes how to “roll up” the graphene sheet to make the
nanotube. The nanotube diameter (d = (n 2 + m 2 + nm) 1/2 ), its density, lattice
structure and the electronic characteristics, such as the conductance, depend only
on the chirality indexes (n, m). A SWNT is considered metallic, with conductivity
showing rectification or ohmic characteristics, if the fraction (n − m)/3 is an
integer value. Otherwise, the nanotube is semiconducting, with variable energy gaps
ranging from a few meV to a few tenths of an eV. In this section we will investigate
the electronic properties of small semiconductor CNTs using high-level accuracy
many-body perturbation theory. In particular, we use the GW approximation, which
assumes that the self-energy of a many-body electron system can be assessed by
keeping only the lowest order term in the expansion of the self-energy in powers of
the screened interaction W . By doing so the self-energy is written as the product of
G, the one-body Green’s function, and W , the screened Coulomb interaction [84].
5.5.1 The GW Method for CNTs
Recently, Umari and coworkers have developed a method for performing accurate
and well-converged GW calculations in large simulation cells based on reduced
basis sets for expressing the polarizability operators [85] and on Lanczos’s chains
for avoiding sums over unoccupied one-particle states [86]. This scheme, which will
be revised in this section, was used for investigating the dependence of electronic
band gaps with respect to the tube diameter for a number of semiconducting singlewall zig-zag CNTs, with diameters ranging from 0.56 to 1.27 nm [11, 12].
Beyond the independent electron approximation, the electronic states in a
strongly interacting system can be described through a quasi-particle picture in
which the behaviour of the bare electron is strongly renormalized by the presence
of the other electrons of the system. On the experimental side, this quasi-particle
behaviour can be probed via direct and inverse photoelectron spectroscopy. On the
theoretical side, MBPT and, in particular, the G 0 W 0 approximation [87] represent
a rather simple, though accurate approach for addressing quasi-particle calculations
on large systems. The G 0 W 0 approach allows one to apply many-body perturbative
corrections to a starting DFT calculation. Within this method, the quasi-particle
energy level E i for the i-th Kohn-Sham state is obtained from the solution of the
following self-consistent one-variable equation:
E i = i + ψ i | c (E i ) |ψ i − ψ i | V xc |ψ i + ψ i | x |ψ i ,
(5.7)
where ψ i is the i-th Kohn-Sham eigenstate, i the eigenenergy, V xc is the exchange
and correlation potential and c and x are the correlation and exchange parts
S. Taioli
graphene, however, the electronic properties of single CNTs depend on their
chirality. The nanotube chirality is defined by the specific and discrete chiral angles
at which graphene sheets are rolled. In particular, the chirality is identified by a
couple of numbers (n, m), which define a vector (C h = na 1 + ma 2 ) in an infinite
graphene sheet that describes how to “roll up” the graphene sheet to make the
nanotube. The nanotube diameter (d = (n 2 + m 2 + nm) 1/2 ), its density, lattice
structure and the electronic characteristics, such as the conductance, depend only
on the chirality indexes (n, m). A SWNT is considered metallic, with conductivity
showing rectification or ohmic characteristics, if the fraction (n − m)/3 is an
integer value. Otherwise, the nanotube is semiconducting, with variable energy gaps
ranging from a few meV to a few tenths of an eV. In this section we will investigate
the electronic properties of small semiconductor CNTs using high-level accuracy
many-body perturbation theory. In particular, we use the GW approximation, which
assumes that the self-energy of a many-body electron system can be assessed by
keeping only the lowest order term in the expansion of the self-energy in powers of
the screened interaction W . By doing so the self-energy is written as the product of
G, the one-body Green’s function, and W , the screened Coulomb interaction [84].
5.5.1 The GW Method for CNTs
Recently, Umari and coworkers have developed a method for performing accurate
and well-converged GW calculations in large simulation cells based on reduced
basis sets for expressing the polarizability operators [85] and on Lanczos’s chains
for avoiding sums over unoccupied one-particle states [86]. This scheme, which will
be revised in this section, was used for investigating the dependence of electronic
band gaps with respect to the tube diameter for a number of semiconducting singlewall zig-zag CNTs, with diameters ranging from 0.56 to 1.27 nm [11, 12].
Beyond the independent electron approximation, the electronic states in a
strongly interacting system can be described through a quasi-particle picture in
which the behaviour of the bare electron is strongly renormalized by the presence
of the other electrons of the system. On the experimental side, this quasi-particle
behaviour can be probed via direct and inverse photoelectron spectroscopy. On the
theoretical side, MBPT and, in particular, the G 0 W 0 approximation [87] represent
a rather simple, though accurate approach for addressing quasi-particle calculations
on large systems. The G 0 W 0 approach allows one to apply many-body perturbative
corrections to a starting DFT calculation. Within this method, the quasi-particle
energy level E i for the i-th Kohn-Sham state is obtained from the solution of the
following self-consistent one-variable equation:
E i = i + ψ i | c (E i ) |ψ i − ψ i | V xc |ψ i + ψ i | x |ψ i ,
(5.7)
where ψ i is the i-th Kohn-Sham eigenstate, i the eigenenergy, V xc is the exchange
and correlation potential and c and x are the correlation and exchange parts