The influence of interlayer bias and crystal field on the electronic characteristics of twisted tri-layer graphene

Twisted van der Waals multilayers have proven to be highly effective in solid-state systems, facilitating the emergence of unique quantum behaviors. By utilizing a real-space tight-binding model, we demonstrate that in twisted trilayer graphene (<i>t</i>-TLG), both localized and dispersive modes can be significantly altered through adjustments in the interlayer bias and crystal field. Interestingly, the interlayer bias results in Dirac crossings above and below the charge neutrality point, alongside several anti-crossings. In contrast, the crystal field creates asymmetry between the inner and outer layers by applying differing electrostatic potentials, which in turn inherently induces an interlayer bias. Our findings indicate that an accurate prediction of the electronic characteristics of <i>t</i>-TLG requires accounting for the effects of both interlayer bias and crystal fields.