• quantum topological transport

Thermoelectricity refers to the phenomenon in which temperature differences are directly converted into electric voltages (and vice-versa) in solid state systems. Thanks to thermoelectric effects, it is possible to realize small solid state thermal machines, which exploit incoming heat fluxes to produce usable work in a very reliable way and without polluting emissions. However, those devices suffer from low efficiency in heat-to-work conversion with respect to their mechanical counterparts. The efficiency of such devices can be quantified by a dimensionless parameter called the figure of merit (ZT), which is completely determined by the transport coefficients. In particular, this parameter is defined as ZT = T σS²/(κ(e) + κ(l)), where σ is the electric conductivity, S the thermopower, κ(e/l) the electronic/lattice thermal conductivity and T the temperature. The figure of merit is of crucial importance because when it tends to infinity, the efficiency of the system tends to the Carnot efficiency. A very promising solution to the efficiency problem of solid state thermal devices is to exploit the quantum phenomena arising in nanoscopic devices. For instance, resonant systems which realize narrow energy filters are expected to exhibit large heat-to-work conversion efficiency, due to the suppressed thermal conductance.
In this thesis we investigate the potential impact in thermoelectricity of topological insulators.This type of materials represents a new state of matter, in which the bulk behaves as an insulator, while the boundaries as conducting states. In 2D systems they are one dimensional edge states and in 3D they are two dimensional surface states. These boundary states are characterized by a linear dispersion in energy (Dirac-like) appearing inside the bulk gap. Moreover, such boundary states have the peculiar characteristic of being topologically protected, which means that they are insensitive to smooth deformations of the Hamiltonian, so that they can not be destroyed by disorder, for example due to the presence of impurities. The presence of disorder lowers the lattice thermal conductivity, while the topological protection of the surface states against disorder implies that the electronic thermoelectric properties of such states remain unchanged with respect to a clean system. Materials that have typically good thermoelectric properties such as Bi₂Te₃, Sb₂Te₃, Bi₂Se₃, have been discovered to be also topological insulators. Despite this interesting properties, there is very little literature about thermoelectric transport in topological insulators, and in most cases only a semiclassical picture is considered. In this thesis we present a theoretical study of thermoelectric quantum transport in 3D topological insulators, in order to better understand this phenomena at the nano-scale and to optimize the efficiency of such systems.
The properties of topological insulators are well described by the topological band theory in the non-interacting electrons approximation, where effective model Hamiltonians for 2D and 3D topological insulators have been derived. In this thesis we consider a 3D topological insulator in a two-terminal set up, where the system is attached to two reservoirs. Through the Landauer-Büttiker scattering theory, we investigated the quantum coherent transport in such systems. Within this theory the electric and heat currents are expressed in terms of the transmission probability for an electron to be transmitted from one terminal to the other. We have started our investigation by studying the thermoelectric behavior of one surface of a 3D topological insulator by means of an analytical model. First, in the presence of one potential barrier, we have seen that the Dirac-like dispersion of the surface states, at normal incidence of the incoming electrons, leads to a constant and perfect transmission with an arbitrary barrier, as we expected (Klein paradox). This is a good feature for the electric conductance, which is at its maximum and constant value. Nevertheless, a constant transmission leads to a zero thermopower and consequently to a zero efficiency. Conversely, at non-normal incidence of the incoming electrons, the transmission probability shows a strong dependence on energy. Precisely it has an oscillating behavior with very high barriers, tunable through the incidence angle and the width of the barrier. In particular we have discovered a dependence of the efficiency on the incidence angle, which is also dependent on temperatures. Specifically, for low temperatures the maximum efficiency is a growing function of the incidence angle. When the angle of incidence is large and the width of the potential barrier small, it is possible to create a very picked transmission which acts as an energy filter and allows very high efficiency, with ZT ∼ 30 (ZT increases also with temperature). We have also calculated the overall transport coefficients summing over all the angles of incidence. We have noticed that in this case the efficiency is lowered with respect to the single mode case, despite of an increased value of the conductances. Moreover, the maximum efficiency is a growing function of the barrier’s width at low temperatures.
Second, we have considered two potential barriers in series. In this case, we analyzed the dependence on the incident angle, width and height of the barriers. In this way we were able to create an energy filter exploiting the resonances energies raising when a second barrier is present. In this case we obtained a high efficiency (ZT ∼ 3), but lower than the previous one barrier case. This is due to the larger number of picks obtained with two barriers with respect to one barrier, which at high temperatures icreases the thermal conductance. When summing over the incident angles we have studied the dependence of the maximum efficiency on the
width of the single barriers and temperature. In particular, at low temperatures the efficiency is a growing function of the width of the single barriers.
As a second step of our investigation we used a more refined full 3D model and calculate numerically the transport quantities on a clean slab of 3D topological insulator in order to corroborate our analytical results. Starting from the continuum model Hamiltonian, we derived a tight binding one through a discretization procedure. With such tight binding Hamiltonian we were able to calculate numerically the scattering matrix by using a numerical toolbox which matches the wave functions of the leads and scatterer. With this effective model we have simulated a realistic slab of topological insulator. We have found that also in this system it is possible to achieve transmission probabilities for the surfaces states strongly dependent on energy, and we have exploited this to create an energy filter introducing two potential barriers. We found high efficiencies which confirms the behavior predicted in the analytical model. Finally, we verified the topological protection of the surfaces states by introducing disorder, finding that even with strong disorder the surfaces states remain virtually unchanged. This gives us confidence that the results obtained for a clean system are also valid in the presence of disorder, which in turn strongly suppress the transport of the phonons in the bulk of the slab.