From phase- to amplitude-fluctuation driven superconductivity in systems with precursor pairing

The change-over from phase- to amplitude-fluctuation-driven superconductivity is examined for a composite system of free electrons (fermions with concentration ${n}_{F})$ and localized electron pairs (hard-core bosons with concentration ${n}_{B})$ as a function of doping\char22{}changing the total concentration of charge carriers ${(n}_{\mathrm{tot}}{=n}_{F}{+2n}_{B}).$ The coupling together of these two subsystems via a charge exchange term induces electron pairing and ultimately superconductivity in the fermionic subsystem. The difference in statistics of the two species of charge carriers has important consequences on the doping mechanism, showing an onset temperature ${T}^{*}$ of incoherent electron pairing in the fermionic subsystem (manifest in form of a pseudogap), which steadily decreases with decreasing ${n}_{\mathrm{tot}}.$ Below ${T}^{*}$ this electron pairing leads, in the normal phase, to electron-pair resonant states (Cooperons) with quasiparticle features which strongly depend on ${n}_{\mathrm{tot}}.$ For high concentrations, where ${n}_{B}\ensuremath{\simeq}0.5,$ correlation effects between the hard-core bosons lead to itinerant Cooperons having a heavy mass ${m}_{p},$ but are long lived. Upon reducing the concentration of charge carriers and consequently ${n}_{B},$ the mass as well as the lifetime of those Cooperons is considerably reduced. As a result, for high values of ${n}_{B},$ a superconducting state below ${T}^{*}$ sets in at a ${T}_{c},$ being controlled by the phase stiffness ${D}_{\ensuremath{\varphi}}={\ensuremath{\Elzxh}}^{2}{n}_{p}{/m}_{p}$ of those Cooperons, where ${n}_{p}$ denotes their density. Upon reducing ${n}_{\mathrm{tot}},$ the phase stiffness steadily increases, and eventually exceeds the pairing energy ${k}_{B}{T}^{*}.$ There, the Cooperons loose their well defined itinerant quasi-particle features and superconductivity gets controlled by amplitude fluctuations. The resulting phase diagram with doping is reminiscent of that of the phase fluctuation scenario for high-${T}_{c}$ superconductivity, except that in our scenario the determinant factors are the mass and the lifetime of the Cooperons rather than their density.