We present a comprehensive investigation of the electronic structures of the $R\mathrm{F}{\mathrm{e}}_2{X}_2$ family ($R$ = Y, La, Lu; $X$ = Ge, Si) using density functional theory (DFT) combined with dynamical mean-field theory (DMFT), with a particular focus on the origin of the flat band observed in $\mathrm{YF}{\mathrm{e}}_2\mathrm{G}{\mathrm{e}}_2$. Our calculations reveal that besides $\mathrm{YF}{\mathrm{e}}_2\mathrm{G}{\mathrm{e}}_2$, only $\mathrm{LuF}{\mathrm{e}}_2\mathrm{G}{\mathrm{e}}_2$ exhibit a nearly flat band around the M point. We demonstrate that, in addition to kinetic frustration and strong electronic correlation effects, hybridization between the $R$ and Fe $d$-orbitals plays a crucial role in the formation of flat bands. By manipulating the crystal structure and doping of $\mathrm{LuF}{\mathrm{e}}_2\mathrm{G}{\mathrm{e}}_2$ to tune the hybridization between Lu and Fe, we successfully reproduce a flat band similar to that in $\mathrm{YF}{\mathrm{e}}_2\mathrm{G}{\mathrm{e}}_2$. The high sensitivity of the flatness of the heavy band in $\mathrm{LuF}{\mathrm{e}}_2\mathrm{G}{\mathrm{e}}_2$ to structural modifications and doping provide us a new strategy for manipulating $d$-electron heavy fermion states. Our findings not only clarify how kinetic frustration, hybridization, and strong electronic correlation effects synergistically drive the emergence of flat bands in $R\mathrm{F}{\mathrm{e}}_2{X}_2$ but also pave the way for exploring new $d$-electron heavy fermion systems.

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• Electronic structure
• First-principles calculations
• Flat bands
• Iron-based superconductors
• Strongly correlated systems
• Dynamical mean field theory