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Magnetic response of twisted bilayer graphene

Simon Becker, Jihoi Kim, Xiaowen Zhu

Abstract

In this article, we analyse the Bistritzer--MacDonald (BM) model (also known as the continuum model) of twisted bilayer graphene (TBG) with an additional external magnetic field. We provide an explicit semiclassical asymptotic expansion of the density of states (DOS) in the limit of strong magnetic fields. The explicit expansion of the DOS enables us to study magnetic response properties such as magnetic oscillations which includes Shubnikov-de Haas and de Haas-van Alphen oscillations as well as the integer quantum Hall effect. In particular, we elucidate the role played by different types of interlayer tunnelings ($AA^{\prime}$/$BB^{\prime}$ vs. $AB^{\prime}$/$BA^{\prime}$) in the study of the DOS, and magnetic properties.

Magnetic response of twisted bilayer graphene

Abstract

In this article, we analyse the Bistritzer--MacDonald (BM) model (also known as the continuum model) of twisted bilayer graphene (TBG) with an additional external magnetic field. We provide an explicit semiclassical asymptotic expansion of the density of states (DOS) in the limit of strong magnetic fields. The explicit expansion of the DOS enables us to study magnetic response properties such as magnetic oscillations which includes Shubnikov-de Haas and de Haas-van Alphen oscillations as well as the integer quantum Hall effect. In particular, we elucidate the role played by different types of interlayer tunnelings (/ vs. /) in the study of the DOS, and magnetic properties.
Paper Structure (22 sections, 24 theorems, 159 equations, 10 figures)

This paper contains 22 sections, 24 theorems, 159 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Introduction of magnetic BM model
  3. Moiré lattices and TBG
  4. Chiral and anti-chiral tunnelings
  5. Magnetic BM model with Adiabatic scaling
  6. Magnetic BM model with Semiclassical Scaling
  7. The chiral and anti-chiral model
  8. Density of states
  9. General properties
  10. Semiclassical expansion of Density of states
  11. First Reduction: Symplectic reduction
  12. Second reduction: Grushin problem
  13. Properties of effective Hamiltonian
  14. Trace formula
  15. Proof of main results
  16. ...and 7 more sections

Key Result

Lemma 3.1

For $f \in C_c^{\infty}({\mathbb R})$ the regularized trace of $f({\mathscr H}^\theta)$ exists, satisfies and depends smoothly on $B \in {\mathbb R}$ and $\theta \in {\mathbb R} \setminus \{0\},$ with Schwartz kernel $f({\mathscr H}^\theta)(x,y)$ of $f({\mathscr H}^\theta).$

Figures (10)

  • Figure 1: On the left: Moiré pattern at twisting angle $\theta = 5^{\circ}$ with single moiré hexagon on the right, with (A=red, B=blue) and (A'=green, B'=black) denote vertices of two sheets of graphene respectively.
  • Figure 2: Modulus of tunneling potentials for various coupling types.
  • Figure 3: Constant magnetic field: On the left, flat bands for chiral model ($\alpha_1=1$); in the middle $(\theta=0)$ and on the right $(\theta=\pi)$ non-flat bands for anti-chiral model, ($\alpha_0=1$).
  • Figure 4: SdH oscillations: Smoothed out DOS $\rho(f_{\mu})$ with $f_{\mu}(x) = e^{-\frac{(x-\mu)^2}{2\sigma^2}}/\sqrt{2\pi}\sigma$ illustrating the oscillatory features. On the left, $B=30$ and on the right $B=50$ for $\sigma=1.$
  • Figure 5: Magnetization and susceptibility for $\beta=4$, $\alpha_i=3/5,$ and chemical potentials $\mu=5$ (left) and $\mu=10$ (right).
  • ...and 5 more figures

Theorems & Definitions (53)

  • Remark 1: Why strong magnetic fields?
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 1: Chiral model
  • Theorem 2: Anti-chiral model
  • Lemma 4.1
  • Remark 2
  • Remark 3
  • ...and 43 more