Electrical conductivity and thermal conductivity are two closely related physical quantities; for example, there is a proportional relationship between them in metals, according to the Wiedemann-Franz law, since valence electrons carry both electric charge and heat energy. Additionally, the thermal conductivity by phonons could also be relevant, as occur in diamond. In general, these two conductivities are relatively easy to measure but quite difficult to model at the atomic scale, since the multiple scattering out of thermodynamic equilibrium should be included.Moreover, for non-periodic hetero structures, the absence of reciprocal space requires new methods for their study.Nowadays, the electronic states in artificial structures is of great importance in the condensed matter physics and materials science, because they introduce many new physical properties essential for industrial applications of atomic-scale devices. In general, the structural disorder of a solid can profoundly modify the localization of its elementary excitations. For example, it is well known that single electronic states are all extended in periodic lattices and exponentially localized in randomly disordered systems of one and two dimensions .
In this talk, we will introduce an original renormalization plus convolution method  developed for the Kubo-Greenwood formula, in order to investigate the frequency-dependent electrical conductivity of quasi periodic systems. This method combines the convolution theorem with the real-space renormalization technique, being able to address multidimensional non-periodic systems with 1024 atoms, including their lattice thermal conductivity .Analytical solutions of the Kubo-Greenwood formula are found for the ballistic DC and AC conductivities in periodic nano wires, where quantized DC conductance steps are observed, in agreement with experimental data . For quasi periodic lattices connected to two semi-infinite periodic leads, the electrical conductivity is calculated by using the renormalization method and the results show that at several frequencies their AC conductivity could be significantly larger than the ballistic one. This fact might be related to the resonant scattering process in quasi periodic systems. Furthermore, calculations made in segmented Fibonacci nano wires reveal that this improvement to the ballistic AC conductivity via quasi periodicity is still present in multidimensional systems as well as at the room temperature .
On the other hand, the direct conversion between thermal and electrical energies by thermoelectric devices have attracted great attention in the last years, and low-dimensional systems seem to be promising candidates for high-performance thermoelectric devices.In particular, segmented nanowires have a band structure by design, which with a properly placed chemical potential by applying a gate voltage could significantly enhance the thermoelectric power . In this work, we present a comparative study of thermoelectricity in periodically and quasi periodically segmented and branched nano wires with macroscopic length by using the renormalization plus convolution method. The results confirm the existence of a maximum thermoelectric figure-of-merit (ZT) around electronic band edges, whose magnitude grows with the reduction of the cross-section area. Finally, we observe a clear enhancement of ZT in quasi periodically segmented nano wires with respect to the periodic ones, mainly due to the reduction of its thermal conductivity by phonons at low temperatures caused by their scattering at the long-range quasi periodically located interfaces .
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