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Doping-induced dimensional crossover and thermopower burst in Nb-doped SrTiO$_3$ superlattices

Using advanced ab-initio calculations, we describe the formation and confinement of a two-dimensional electron gas in short-period ($\simeq$4 nm) Nb-doped SrTiO$_3$ superlattices as function of Nb doping. We predict complete two-dimensional confinement for doping concentrations higher than 70%. In agreement with previous observations, we find a large thermopower enhancement at room temperature. However, this effect is primarily determined by dilution of the mobile charge over a multitude of weakly occupied bands. As a general rule, we conclude that thermopower in similar heterostructures will be more enhanced by weak, rathern than tight spatial confinement.