Heatmaps of NNN-Driven Asymmetry in Graphene

We investigate electron–hole asymmetry in monolayer graphene using a minimal π-band tight-binding model that adds next-nearest-neighbor (NNN) hopping to the standard nearest-neighbor Hamiltonian. While NN hopping alone yields particle–hole symmetry and a Dirac crossing at charge neutrality, a finite NNN term preserves the gapless crossing but shifts the neutrality point and skews band curvatures across the Brillouin zone. We quantify these effects with three complementary outputs: (i) band overlays along Γ–K–M–Γ that show the Dirac crossing moving rigidly with the NNN term and the loss of spectral mirror symmetry; and (ii) a Brillouin-zone "asymmetry map," defined as the sum of the conduction and valence energies, which visualizes a six-fold pattern tied to lattice geometry. We extract a simple "asymmetry score" from the map and track the Dirac energy shift versus the NNN parameter. The results provide a clean baseline for interpreting ARPES in realistically asymmetric graphene devices.