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Two-dimensional (2D) lattices composed exclusively of pentagons represent anexceptional structure of materials correlated to the famous pentagonal tilingproblem in mathematics, but their $\pi$-conjugation and the related electronicproperties have never been reported. Here, we propose a tight-binding (TB)model for a 2D Cairo pentagonal lattice and demonstrate that $p$-$d$$\pi$-conjugation in the unique framework leads to intriguing properties, suchas an intrinsic direct band gap, ultra-high carrier mobility and even slantDirac cones. On the basis of first-principles calculations, we predict acandidate material, 2D penta-NiP$_2$ monolayer, derivated from bulk NiP$_2$crystal, to realize the predictions of the TB model. It has ultra-high carriermobility ($\sim$$10^5-10^6$ $cm^2V^{-1}s^{-1}$) comparable to that of grapheneand an intrinsic direct band gap of 0.818 eV, which are long desired forhigh-speed electronic devices. The stability and possible synthetic routes ofpenta-NiP$_2$ monolayer are also discussed.

Electronic properties of a $\pi$-conjugated Cairo pentagonal lattice: Direct band gap, ultrahigh carrier mobility and slant Dirac cones