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Coupling of the A_{1g} As-phonon to magnetism in iron pnictides

Charge, spin and lattice degrees of freedom are strongly entangled in iron superconductors. A neat consequence of this entanglement is the behavior of the A_{1g} As-phonon resonance in the different polarization symmetries of Raman spectroscopy when undergoing the magneto-structural transition. In this work we show that the observed behavior could be a direct consequence of the coupling of the phonons with the electronic excitations in the anisotropic magnetic state. We discuss this scenario within a five orbital tight-binding model coupled to phonons via the dependence of the Slater-Koster parameters on the As position. We identify two qualitatively different channels of the electron-phonon interaction: a geometrical one related to the Fe-As-Fe angle and another one associated with the modification upon As displacement of the Fe-As energy integrals pdsigma and pdpi. While both mechanisms result in a finite B_{1g} response, the behavior of the phonon intensity in the A_{1g} and B_{1g} Raman polarization geometries is qualitatively different when the coupling is driven by the angle or by the energy integral dependence. We discuss our results in view of the experimental reports.