# Magnetic Field-Controlled Electrical Conductivity in AA Bilayer Graphene

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalization of Hubbard Model for AA Bilayer Graphene

## 3. The Green-Kubo Formalism and Electrical Conductivity in AA Bilayer Graphene

#### 3.1. The Electric Current Operator beyond Dirac Approximation

#### 3.2. The Polarization Function and Electrical Conductivity

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BLG | Bilayer Graphene |

## Appendix A. Green’s Functions and Wick’s Average

## References

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**Figure 1.**Illustration of the AA-stacked BLG system exposed to the external magnetic field B (see the black dashed arrow in the picture) in the direction of the z-axis. The intra-layer and inter-layer hopping amplitudes are shown, and the sublattice notation is provided in each layer (see the sublattice sites ${A}_{1}$ and ${B}_{1}$ in the bottom layer $\ell =1$ and ${A}_{2}$ and ${B}_{2}$ in the top layer $\ell =2$). The electric field potential V is applied to the system.

**Figure 2.**The x-component of the real part of conductivity function as a function of the excitation energy parameter E (in units of ${\gamma}_{0}$). We considered the case of partial-filling in the layers and the inverse filling coefficient is set at the value $\kappa =1$. Different values of the external magnetic field have been considered, from zero (see in panel (

**a**)), up to very high value (see in panel (

**c**)). The separation of the conductivity function, for different spin directions, is well described in panel (

**b**). The inter-layer Coulomb interaction parameters is fixed at the value $W=2{\gamma}_{0}$ and external gate potential at the value $V=2{\gamma}_{0}$. The zero temperature limit is considered.

**Figure 3.**The x-component of the real part of conductivity function as a function of the excitation energy parameter E (in units of ${\gamma}_{0}$). We considered the case of half-filling in the layers and the inverse filling coefficient is set at the value $\kappa =0.5$. Different values of the external magnetic field have been considered, from zero up to very high value (see in panel (

**a**–

**c**)). We see that there is no splitting of the conductivity function due to different spin orientations. The inter-layer Coulomb interaction parameters is fixed at the value $W=2{\gamma}_{0}$ and external gate potential at the value $V=2{\gamma}_{0}$. The zero temperature limit is considered.

**Figure 4.**The x-component of the real part of conductivity function as a function of the excitation energy parameter E (in units of ${\gamma}_{0}$).

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**MDPI and ACS Style**

Apinyan, V.; Kopeć, T.
Magnetic Field-Controlled Electrical Conductivity in AA Bilayer Graphene. *C* **2023**, *9*, 42.
https://doi.org/10.3390/c9020042

**AMA Style**

Apinyan V, Kopeć T.
Magnetic Field-Controlled Electrical Conductivity in AA Bilayer Graphene. *C*. 2023; 9(2):42.
https://doi.org/10.3390/c9020042

**Chicago/Turabian Style**

Apinyan, Vardan, and Tadeusz Kopeć.
2023. "Magnetic Field-Controlled Electrical Conductivity in AA Bilayer Graphene" *C* 9, no. 2: 42.
https://doi.org/10.3390/c9020042