Excitonic insulators are semi-metals in which excitons, electron-hole pairs bound by the Coulomb interaction, condense at low temperatures. Consequently, the energy gap opens in the band structure and unusual transport properties arise. This state of matter was theoretically predicted in the 1960s. Exciton condensation was experimentally confirmed in the 1990s in a special quantum Hall regime in heterostructures, but an unambiguous confirmation of this state in bulk materials is still missing. Recently, candidate materials for excitonic insulators were proposed, e.g. Ta$_2$NiSe$_5$; however, the mechanism of their phase transition to the insulating phase is unclear. Commonly, such questions are addressed by measuring transport properties, which convey the system's response to external perturbations.
In the master thesis, we are interested in thermoelectric transport and, specifically, the Seebeck coefficient. It expresses the voltage induced between ends of a sample due to a temperature gradient. In the first part, we describe a general theory of electronic transport in crystals. Semi-classically, the description is formulated within Boltzmann's theory, while quantum-mechanically, the main tool is Kubo's linear response theory. In the second part, we study a minimal model of the excitonic phase transition in a one-dimensional electron gas within the mean-field approximation, and its Seebeck coefficient. We find that the condensation of excitons can give rise to qualitatively different band structures and temperature dependence of the Seebeck coefficient. An important result is that scattering notably alters thermoelectric transport at low temperatures, which is why the semi-classical description is incomplete. In the third part, we study Ta$_2$NiSe$_5$ within the tight-binding approximation and the mean-field approximation, and numerically evaluate its Seebeck coefficient using both Boltzmann's and Kubo's theory. We compare our results to experimental data and find qualitative agreement at low temperatures. We also identify inconsistencies and propose how they could be resolved in future work.
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