Twistronics versus straintronics in twisted bilayers of graphene and transition metal dichalcogenides

Several numerical studies have shown that the electronic properties of\ntwisted bilayers of graphene (TBLG) and transition metal dichalcogenides (TMDs)\nare tunable by strain engineering of the stacking layers. In particular, the\nflatness of the low-energy moir\\'e bands of the rigid and the relaxed TBLG was\nfound to be, substantially, sensitive to the strain. However, to the best of\nour knowledge, there are no full analytical calculations of the effect of\nstrain on such bands. We derive, based on the continuum model of moir\\'e flat\nbands, the low-energy Hamiltonian of twisted homobilayers of graphene and TMDs\nunder strain at small twist angles. We obtain the analytical expressions of the\nstrain-renormalized Dirac velocities and explain the role of strain in the\nemergence of the flat bands. We discuss how strain could correct the twist\nangles and bring them closer to the magic angle $\\theta_m\\sim1.05^{\\circ}$ of\nTBLG and how it may reduce the widths of the lowest-energy bands at charge\nneutrality of the twisted homobilayer of TMDs. The analytical results are\ncompared with numerical and experimental findings and also with our numerical\ncalculations based on the continuum model.\n