Robust cross-chain surface interstitial electronic states and doping-enhanced superconductivity in monolayer $M_2$N ($M$= Ti, Zr, Hf) electrides

The exploration of electrides holds great promise for advancing both fundamental physics and chemistry, owing to their unique characteristics arising from loosely bound interstitial anionic electrons. Here we report a class of cross-chain electrides, distinguished by two distinct anionic electron subchannels forming alternating chains in real space. Through structural symmetry analysis and first-principles calculations, we identify two-dimensional $M_2$N ($M$ = Ti, Zr, Hf) materials as prototypical systems exhibiting these unique features. The anionic electron channels on the upper and lower surfaces of these materials display a vertically alternating pattern, with their projected bands revealing momentum-dependent splitting behavior in the reciprocal space, protected by a crystal symmetry operation $\mathcal{O}$. Notably, the cross-chain electride characteristic in the $M_2$N monoalyers is independent of the layer number and remains robust on the upper and lower surfaces of layered structures, presenting a pronounced and robust surface interstitial electronic state. Additionally, we have explored the superconductivity of these systems, and found that both Ti$_2$N and Zr$_2$N are intrinsic superconductors with superconducting transition temperatures below 1.0 K. Further results show that appropriate hole doping can significantly enhance their superconducting transition temperatures and can induce the Hf$_2$N monolayer to exhibit superconductivity. Our findings provide valuable insights into the design and tuning of novel electrides with enhanced superconducting properties, offering another pathway for deeply understanding the interplay between electride behavior and superconductivity in novel materials.