# Effect of asymmetric Fermi velocity on trigonally warped spectrum of bilayer graphene

## TL;DR

This paper derives an effective Hamiltonian for bilayer graphene with asymmetric Fermi velocities and explores how this asymmetry affects its band structure, revealing a tunable indirect band gap and electron-hole asymmetry.

## Contribution

It introduces a new effective Hamiltonian accounting for Fermi velocity asymmetry and analyzes its impact on the band structure of trigonally warped bilayer graphene.

## Key findings

- Fermi velocity asymmetry preserves threefold symmetry in low-energy band structure.
- An indirect, tunable band gap arises due to velocity asymmetry.
- Velocity asymmetry causes significant electron-hole asymmetry around trigonal pockets.

## Abstract

We derive an effective Hamiltonian at low energies for bilayer graphene when Fermi velocity manufactured on each layer is different of the velocity measured in pristine graphene. Based on the effective Hamiltonian, we investigate the influence of Fermi velocity asymmetry on the band structure of trigonally warped bilayer graphene in the presence of interlayer applied bias. In this case, the Fermi line at low energies is still preserved its threefold rotational symmetry appearing as the three pockets. Furthermore, the interlayer asymmetry in Fermi velocities leads to an indirect band gap which its value is tunable by the velocity ratio of the top to bottom layer. It is also found that one of the origins for emerging the electron-hole asymmetry in the band structure, is the velocity asymmetry which is large around the trigonal pockets.