Quantum tunneling and defect-induced transport modulation in twisted bilayer graphene superlattices

TL;DR

This work addresses quantum tunneling through a finite periodic superlattice in twisted bilayer graphene (TBG) with rectangular barriers and a single defect, using a low-energy continuum model. The authors derive an effective Hamiltonian $H^{eff}$ and a periodic potential profile $U_j(x)$ with barrier height and defect height, solving for multiregion states under $H_j=H^{eff}+U_j(x)\,\mathbb{I}_2$ and matching boundary conditions to obtain transmission spectra. They report that transmission exhibits miniband-like oscillations with four resonance series and $N-1$ minigaps, whose number, depth, and positions are tunable by the twist angle $\theta$, barrier number $N$, barrier/well widths $d_B$, $d_w$, and defect parameters, including defect-induced in-gap tunneling states whose energy shifts with $d_w$. At low energy, normal-incidence transmission is near-perfect independent of $\theta$ and $N$, while at higher energy transmission becomes anisotropic due to Dirac-cone separation; defects at small $\theta$ can suppress Klein tunneling, offering an extra knob for transport control. The results illuminate how moiré-induced anisotropy and finite-periodicity in TBG combine to shape tunneling, with potential applications in twist-tunable nanoelectronic and quantum devices, such as wave-vector filtering and angle-selective transport.

Abstract

We investigate quantum tunneling of charge carriers through a periodic superlattice in twisted bilayer graphene (TBG) with rectangular potential barriers, including the presence of a defect, using a low-energy continuum model. Transmission probabilities are numerically analyzed depending on the parameters of the problem, highlighting the roles of twist angle, number of barriers, barrier geometry, and the presence of a defect barrier within the superlattice. Our numerical results reveal that transmission is highly sensitive to these parameters: reducing the twist angle changes the number, depth, and position of transmission gaps and resonance peaks. The presence of defect affects the transmission, leading to the appearance of tunneling states inside transmission gaps with energy position can be tuned by the well width. At low incident energy, the transmission for normally incident electrons is perfect or nearly perfect, independent of the twist angle and the number of barriers. However, at large incident energy, the transmission becomes distinctly anisotropic, reflecting the separation of Dirac cones induced by twist angle variations. The presence of defects, particularly at smaller twist angles, provides additional control of tunneling behavior, allowing complete suppression of Klein tunneling under certain conditions. These findings extend the established understanding of miniband transport in periodic graphene systems and open new possibilities for twist-tunable nanoelectronic and quantum devices.

Figures (6)

• Figure 1: A schematic diagram of a quasiparticle coming with an incident energy $E$ to a periodic potential barrier of TBG superlattices whith height of $U_{j}=E+\Delta U_{0}$ and the period of the superlattice is $d=d_{B}+d_{w}$. Here, $d_{B}$ and $d_{W}$ denote the barrier and well widths, respectively. We assume that the single barrier has a rectangular shape and is infinite along the $y$-direction. $k_{x}$ has a perpendicular direction to the barrier.
• Figure 2: Transmission probability for normally incident electrons as a function of the incident energy through different numbers of symmetrical potential barriers $U_{N}=E+\Delta U_{0}$ where $\Delta U_{d}=\Delta U_{0}=0.3eV$ (here, $E$ is the incident energy of the electron). The remaining parameters are the twist angle $\theta=3.89^{\circ}$, the barrier width $d_{B}=30\;nm$, and the well width width $d_{w}=25\;nm$.
• Figure 3: (color online). Quantum tunneling in a TBG superlattice for normally incident electrons. Transmission probability for normally incident electrons as a function of the incident energy, through a periodic structure with different number of barriers $N$. The curves with different colors correspond to different values of $\theta$. The remaining parameters are $\Delta U_{d}=\Delta U_{0}=0.3eV$, the barrier width $d_{B}=30\;nm$, and the well width $d_{w}=25\;nm$.
• Figure 4: (color online). Quantum tunneling in a TBG superlattices for normally incident electrons. Transmission probability for normally incident electrons as a function of the incident energy, through a TBG superlattice for (a),(b) periodic structure without defect; (c),(d) periodic structure with defect where $\Delta U_{d}>\Delta U_{0}$; and (e),(f) periodic structure with defect where $\Delta U_{d}<\Delta U_{0}$. The curves with different colors correspond to different values of the well width $d_{w}$. The left and right panels correspond to the case of $N=4$ and $N=9$, respectively. The remaining parameters are $\Delta U_{0}=0.3eV$, the twist angle $\theta=3^{\circ}$, and the barrier width $d_{B}=30\;nm$.
• Figure 5: (color online). Quantum tunneling in a TBG superlattice for low incident energy. Transmission probability for incident electrons as a function of the incident angle $\varphi$ through a periodic structure of TBG superlattices without defect for different number of barriers $N$. The curves with different colors correspond to different values of $\theta$. The remaining parameters are the barrier width $d_{B}=30\;nm$, and the well width $d_{w}=25\;nm$.