Anharmonic softening of Raman active phonons in iron-pnictides: Estimating the Fe isotope effect due to anharmonic expansion

Abstract

We present Raman measurements on the iron-pnictide superconductors ${\text{CeFeAsO}}_{1\ensuremath{-}x}{\text{F}}_{x}$ and ${\text{NdFeAsO}}_{1\ensuremath{-}x}{\text{F}}_{x}$. Modeling the Fe-As plane in terms of harmonic and a cubic anharmonic Fe-As interactions, we calculate the temperature dependence of the energy and lifetime of the Raman active $\text{Fe}\text{ }{B}_{1g}$ mode and fit to the observed energy shift. The shifts and lifetimes are in good agreement with those measured also in other Raman studies which demonstrate that the phonon spectrum, at least at small wave numbers, is well represented by phonon-phonon interactions without any significant electronic contribution. Even at zero temperature there is a non-negligent effect of interactions on the phonon energy, which for the $\text{Fe}\text{ }{B}_{1g}$ mode corresponds to $6\text{ }{\text{cm}}^{\ensuremath{-}1}$ or 3% of the total energy of the mode. We also estimate the anharmonic expansion from Fe $(56\ensuremath{\rightarrow}54)$ isotope substitution to $\ensuremath{\Delta}a\ensuremath{\approx}5.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\text{ }\text{\AA{}}$ and $\ensuremath{\Delta}{d}_{\text{Fe-As}}\ensuremath{\approx}2.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\text{ }\text{\AA{}}$ and the shift of harmonic zero-point fluctuations of bond lengths $⟨\ensuremath{\Delta}{x}^{2}⟩\ensuremath{\lesssim}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\text{ }{\text{\AA{}}}^{2}$, giving a total relative average decrease in electronic hopping integrals of $|\ensuremath{\delta}t|/t\ensuremath{\lesssim}2.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. For a nonphonon-mediated weak-coupling superconductor this gives an isotope exponent $\ensuremath{\alpha}\ensuremath{\sim}{10}^{\ensuremath{-}2}$. The results pose a serious challenge for any theory of superconductivity in the pnictides that does not include electron-phonon interactions to produce a sizable Fe isotope effect.