In-plane magnetoelectric response in bilayer graphene

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 quasidegenerate 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.

Figure 1

Top view on the crystal lattice of a Bernal-stacked graphene bilayer. Open circles indicate sites on the dimer sublattices $A$ and $A^{\prime }$, while the blue (light blue) closed circles are sites on the nondimer sublattice $B$ ($B^{\prime }$).

Figure 2

Side view of the crystal lattice of a Bernal-stacked graphene bilayer. The red (blue) spheres depict the atoms of the sublattices $A$ and $A^{\prime }$ ($B$ and $B^{\prime }$). The yellow and green connecting lines represent the intra- and interlayer couplings, respectively.

Figure 3

Effect of an in-plane magnetic field $\mathbf{B}=B\widehat{\mathbf{x}}$ on the band structure of bilayer graphene. Results shown in panel (a) [(b), (c)] have been obtained by diagonalizing the tight-binding Hamiltonian at the $\mathbf{K}$ valley, Eq. (20), for $B=0$ [$B=1000\mathrm{T}$ and assuming ${\gamma }_3={\gamma }_4={\gamma }_0^{\prime }=0$, $B=1000\mathrm{T}$ without any approximation]. Numerical values used for band-structure parameters are listed in Table 1.

Figure 4

Low-energy dispersion at the $\mathbf{K}$ valley computed for $\mathcal{E}=\mathit{0}$ and $\mathbf{B}=B\widehat{\mathbf{x}}$ with $B=300\mathrm{T}$ using Eq. (20) (yellow), Eq. (30) (red-dotted), and Eq. (30) including the parabolic terms from Eq. (B1) (blue-dashed). Numerical values used for band-structure parameters are listed in Table 1.

Figure 5

Illustration of the valley-dependent optical absorption for different chemical potentials $\tilde{\mu }$ due to the axionic energy shift ${\mathrm{\Delta }}_{\text{ax}}$. The minimum transition frequencies ${\omega }_{\pm }$ are (a) identical for $\tilde{\mu }=0$ and (b) distinct for $\tilde{\mu }\ne 0$ (here $\tilde{\mu }>0$).

Figure 6

Low-energy optical conductivity spectrum in terms of ${\sigma }_0=2e^2/h$ at zero temperature in (a) without static electric and magnetic fields and in (b), (c) for parallel fields (here, $\mathcal{E}\parallel \mathbf{B}\parallel \widehat{\mathbf{y}}$ with ${\mathcal{E}}_y=0.05\mathrm{V}/\mathrm{nm}$ and $B_y=100\mathrm{T}$). The different colors correspond to different magnitudes of the chemical potential $\tilde{\mu }$ in units of ${10}^{-3}{\gamma }_1$. The solid lines correspond to the exact numerical calculation. In (c), the plots for the lowest three chemical potentials are enlarged and compared to the simplified model where ${\gamma }_3={\gamma }_4={\gamma }_0^{\prime }=0$ (dashed lines). Numerical values used for band-structure parameters are listed in Table 1.

Figure 7

(a) Optical conductivity in units of ${\sigma }_0=2e^2/h$ obtained at zero temperature including the corrections due to the carrier drift for different relaxation times ${\tau }_{\mathrm{tr}}$. The chemical potential at $\mathbf{k}=\mathit{0}$ is selected as $\tilde{\mu }\left(0\right)=0.1{\gamma }_1$. The fields are chosen analogously to Fig. 6, i.e., $\mathcal{E}\parallel \mathbf{B}\parallel \widehat{\mathbf{y}}$ with ${\mathcal{E}}_y=0.05\mathrm{V}/\mathrm{nm}$ and $B_y=100\mathrm{T}$. The dashed curve refers to the case without drift current. Panel (b) shows the corresponding energy dispersion at the $\pm \mathbf{K}$ valleys for $k_x=0$. The linear plots with different slope illustrate the $\mathbf{k}$-dependent chemical potential $\tilde{\mu }\left(\mathbf{k}\right)$, Eq. (38), for the different relaxation times. Numerical values used for band-structure parameters are listed in Table 1.