Band structure and Klein paradox for a pn junction in ABCA-tetralayer graphene

We investigate the band structure of ABCA-tetralayer graphene (ABCA-TTLG) subjected to an external potential $V$ applied between top and bottom layers. Using the tight-binding model, including the nearest $t$ and next-nearest-neighbor $t'$ hopping, low-energy model and two-band approximation model we study the band structure variation along the lines $\Gamma-M-K-\Gamma$ in the first Brillouin zone, electronic band gap near Dirac point $K$ and transmission properties, respectively. Our results reveal that ABCA-TTLG exhibits markedly different properties as functions of $t'$ and $V$. We show that the hopping parameter $t'$ changes the energy dispersion, the position of $K$ and breaks sublattice symmetries. A sizable band gap is created at $K$, which could be opened and controlled by the applied potential $V$. This gives rise to 1D-like van Hove singularities (VHS) in the density of states (DOS). We study the relevance of the skew hopping parameters $\gamma_3$ and $\gamma_4$ to these properties and show that for energies $E\gtrsim6$meV their effects are negligible. Our results are numerically discussed and compared with the literature.