# In-plane magnetoelectric response in bilayer graphene

## TL;DR

This paper investigates the in-plane magnetoelectric response of bilayer graphene, linking it to valleytronics, and derives a comprehensive low-energy Hamiltonian including key magnetoelectric terms.

## Contribution

It provides a detailed derivation of the low-energy Hamiltonian for bilayer graphene with in-plane fields, revealing an axion-like term and anisotropic corrections related to interlayer hopping.

## Key findings

- Confirmed the existence of an axion-type pseudoscalar magnetoelectric term.
- Identified anisotropic corrections related to skew interlayer hopping parameter γ4.
- Suggested optical conductivity features to detect magnetoelectric effects.

## Abstract

A graphene bilayer shows an unusual magnetoelectric response whose magnitude is controlled by the valley-isospin density, making it possible to link magnetoelectric behavior to valleytronics. Complementary to previous studies, we consider the effect of static homogeneous electric and magnetic fields that are oriented parallel to the bilayer's plane. Starting from a tight-binding description and using quasi-degenerate perturbation theory, the low-energy Hamiltonian is derived including all relevant magnetoelectric terms whose prefactors are expressed in terms of tight-binding parameters. We confirm the existence of an expected axion-type pseudoscalar term, which turns out to have the same sign and about twice the magnitude of the previously obtained out-of-plane counterpart. Additionally, small anisotropic corrections to the magnetoelectric tensor are found that are fundamentally related to the skew interlayer hopping parameter $\gamma_4$. We discuss possible ways to identify magnetoelectric effects by distinctive features in the optical conductivity.