Thickness dependence of the superconducting critical temperature in heavily doped Si:B epilayers

Abstract

We report on the superconducting properties of a series of heavily doped Si:B epilayers grown by gas immersion laser doping with boron content (${n}_{B}$) ranging from $\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}{10}^{20}$ cm${}^{\ensuremath{-}3}$ to $\ensuremath{\sim}6\ifmmode\times\else\texttimes\fi{}{10}^{21}$cm${}^{\ensuremath{-}3}$ and thickness ($d$) varying between $\ensuremath{\sim}20$ nm and $\ensuremath{\sim}210$ nm. We show that superconductivity is only observed for ${n}_{B}$ values exceeding a threshold value (${n}_{c,S}$) which scales as ${n}_{c,S}\ensuremath{\propto}1/d$. The critical temperature (${T}_{c}$) then rapidly increases with ${n}_{B}$, largely exceeding the theoretical values which can be estimated by introducing the electron-phonon coupling constant (${\ensuremath{\lambda}}_{e\ensuremath{-}ph}$) deduced from ab initio calculations into the McMillan equation. Surprisingly ${T}_{c}({n}_{B},d)$ is fully determined by the boron dose (${n}_{B}\ifmmode\times\else\texttimes\fi{}d$) and can be well approximated by a simple ${T}_{c}({n}_{B},d)\ensuremath{\approx}{T}_{c,0}[1\ensuremath{-}A/({n}_{B}.d)]$ law, with ${T}_{c,0}\ensuremath{\sim}750$ mK and $A\ensuremath{\sim}8(\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{15}$ cm${}^{\ensuremath{-}2}$.