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Suppression of the Berezinskii-Kosterlitz-Thouless and Quantum Phase Transitions in 2D Superconductors by Finite Size Effects

We perform a detailed finite-size scaling analysis of the sheet resistance in Bi-films and the LaAlO$_{3}$/SrTiO$_{3}$ interface in the presence and absence of a magnetic field applied perpendicular to the system. Our main aim is to explore the occurrence of Berezinskii-Kosterlitz-Thouless (BKT) and quantum phase transition behavior in the presence of limited size, stemming from the finite extent of the homogeneous domains or the magnetic field. Moreover we explore the implications thereof. Above an extrapolated BKT transition temperature, modulated by the thickness $d$, gate voltage $%V_{g}$ or magnetic field $H$, we identify a temperature range where BKT behavior occurs. Its range is controlled by the relevant limiting lengths,which are set by the extent of the homogeneous domains or the magnetic field. The extrapolated BKT transition lines $T_{c}\left(d,V_{g},H\right) $ uncover compatibility with the occurrence of a quantum phase transition where $T_{c}\left( d_{c},V_{gc},H_{c}\right) =0$. However, an essential implication of the respective limiting length is that the extrapolated phase transition lines $T_{c}\left( d,V_{g},H\right) $ are unattainable. Consequently, given a finite limiting length, BKT and quantum phase transitions do not occur. Nevertheless, BKT and quantum critical behavior is observable, controlled by the extent of the relevant limiting length. Additional results and implications include: the magnetic field induced finite size effect generates a flattening out of the sheet resistance in the $T\rightarrow 0$ limit, while in zero field it exhibits a characteristic temperature dependence and vanishes at $T=0$ only. The former prediction is confirmed in both, the Bi-films and the LaAlO$_{3}$/SrTiO$_{3}$ interface, as well as in previous studies. The latter is consistent with the LaAlO$_{3}$/SrTiO$_{3}$ interface data, while the Bi-films exhibit a flattening out.