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approach for transport computations is based on the density functional tight-binding
(DFTB) method [8], extended to the non-equilibrium Green’s functions (NEGF) for the
self-consistent computation of charge density and electronic transport [9].
The gDFTB method allows a nearly first-principle treatment of systems comprising
a large number of atoms. The Green’s function technique enables the computation of the
tunnelingcurrent flowingbetweentwocontactsinamannerconsistentwiththeopenbound-
ary conditions that naturally arise in transport problems. On the other hand, the NEGF
formalism allows to compute the charge density consistently with the non-equilibrium
conditions in which a molecular device is driven when biased by an external field.
The key ingredient of the self-consistent loop is the solution of the Hartree potential
needed in the density functional Hamiltonian. The Hartree potential is calculated by solv-
ing the three-dimensional Poisson’s equation (with appropriate boundary conditions),
for the corresponding non-equilibrium charge density computed via the NEGF formal-
ism. The Green’s function method also allows extensions to include electron–phonon
and many-body corrections of the electron–electron interaction.
A full description of our methodology is given in the first sections. Applications to the
computation of conductance of various molecular systems are also shown, particularly
applications to IETS of octane-thiols and heat dissipation in dithio-benzene. We also
present how to include electron–electron interactions beyond DFT, using the well-known
GW approach. In the last section a detailed study of carbon nanotube field-effect devices
is presented.
2. DFTB as a semi ab-initio approach
The density functional–based tight-binding formalism (DFTB) has been described in
detail in many articles and reviews [8]. All matrix elements and orbital wavefunctions
are derived from density-functional calculations. The advantage of the method relies on
the use of a small basis set and the restriction to two center integrals, allowing extensive
use of look-up tables. What distinguishes our approach from empirical methods is the
explicit calculation of the basis wavefunctions, which allows deeper physical insights
and better control of the approximations used. The method solves the Kohn–Sham
equations self-consistently using a Mulliken charge projection [10].
In the traditional DFTB code a minimal basis set of atomic orbitals is used in order to
reduce the matrix dimensions for diagonalization speed-up. This approach has proved to
give transferable and accurate interaction potentials and the numerical efficiency of the
method allows molecular dynamic simulations of large super-cells, containing several
hundreds of atoms, particularly suitable to study the electronic properties and dynamics
of large mesoscopic systems and organic molecules such as CNTs, DNA strands or
adsorbates on surfaces, semiconducting heterostructure, etc., see [11].
We briefly describe here the self-consistent DFTB method. The method is a develop-
ment of the idea first introduced by Foulkes, where the electronic density is expanded as
a sum of a reference density, n
0
r, (that can be chosen as the superposition of neutral
atomic densities) and a deviation, nr, such that nr =n
0
r+nr. The total energy
of the system can be described, up to second order in the local density fluctuations, as:
E
tot
n =
k
n
k
<
k
H
0
k
> +E
rep
n
0
+E
2
n (1)