Linear scaling relation between two-dimensional massless Dirac fermion Fermi velocity and Fe-As bond length in iron arsenide superconductor systems

Two-dimensional (2D) massless Dirac fermions (MDF), which represent a type of quasi-particles with linear energy-momentum dispersions only in 2D momentum space, provide a fertile ground for realizing novel quantum phenomena. However, 2D MDF were seldom observed in the superconducting bulk states of 3D materials. Furthermore, as a cornerstone for accurately tuning the quantum phenomena based on 2D MDF, a quantitative relationship between 2D MDF and a structural parameter has rarely been revealed so far. Here, we report magneto-infrared spectroscopy studies of the iron-arsenide-superconductor systems NaFeAs and $A\mathrm{Fe_2As_2} (A = \mathrm{Ca, Ba})$ at temperature $T \sim 4.2$ K and at magnetic fields ($B$) up to 17.5 T. Our results demonstrate the existence of 2D MDF in the superconducting bulk state of NaFeAs. Moreover, the 2D-MDF Fermi velocities in NaFeAs and $A\mathrm{Fe_2As_2} (A = \mathrm{Ca, Ba})$, which are extracted from the slopes of the linear $\sqrt{B}$ dependences of the Landau-level transition energies, scale linearly with the Fe-As bond lengths. The linear scaling between the 2D-MDF Fermi velocities and the Fe-As bond lengths is supported by (i) the linear relationship between the square root of the effective mass of the $d_{xy}$ electrons and the Fe-As bond length and (ii) the linear dependence of the square root of the calculated tight-binding hopping energy on the Fe-As bond length. Our results open up new avenues for exploring and tuning novel quantum phenomena based on 2D MDF in the superconducting bulk states of 3D materials.