Moiré circuits: Engineering magic-angle behavior

Moiré superlattices in the twisted bilayer graphene provide an unprecedented platform to investigate a wide range of exotic quantum phenomena. Recently, the twist degree of freedom has been introduced into various classical wave systems, giving rise to new ideas for the wave control. The question is whether twistronics and moiré physics can be extended to electronics with potential applications in the twist-enabled signal processing. Here, we demonstrate both in theory and experiment that lots of fascinating moiré physics can be engineered using electric circuits with extremely high degrees of freedom. By suitably designing the interlayer coupling and biasing of one sublattice for the twisted bilayer circuit, the low-energy flat bands with large band gaps away from other states can be realized at various twist angles. Based on the moiré circuit with a fixed twist angle, we experimentally demonstrate the effect of band narrowing as well as the localization of electric energy when a magic value of the interlayer coupling is applied. Furthermore, the topological edge states, which originate from the moiré potential induced pseudomagnetic field, are also observed. Our findings suggest a flexible platform to study twistronics beyond natural materials and other classical wave systems, and may have potential applications in the field of intergraded circuit design.

Figure 1

(a) The moiré superlattice with a commensurate rotation angle being $\theta =5.{09}^{\circ }\left(m=6,n=7\right)$. The pink parallelogram marks the moiré unit cell. (b) The enlarged view of moiré circuits around red stars in Fig. 1. Eight insets present the grounding of different circuit nodes.

Figure 2

(a) Band structures of moiré circuits with $C_{\mathrm{inter}}=0.2C_{\mathrm{intra}}$, $C_{\mathrm{intra}}$, and $2C_{\mathrm{intra}}$. (b) The relationship between the low-energy bandwidth (and band gap) and the ratio of $C_{\mathrm{inter}}/C_{\mathrm{intra}}$ of the biased and unbiased moiré circuits. (c) The blue (red) line presents the enlarged view of band dispersion for the biased (unbiased) circuit with $C_{\mathrm{inter}}=C_{\mathrm{intra}}$. (d), (g) Numerical results of the normalized bulk and edge impedances of the biased $6\times 6$ moiré circuit with $C_{\mathrm{inter}}/C_{\mathrm{intra}}=1$. (e), (f) The dispersion of the biased 1D moiré-ribbon circuit with $C_{\mathrm{inter}}/C_{\mathrm{intra}}=0.2$ and 1. (h), (i) The simulated impedances of a bulk and a top edge node in the biased 1D moiré-ribbon circuit with $C_{\mathrm{inter}}/C_{\mathrm{intra}}=1$.

Figure 3

(a) The top chart displays the photograph image of the moiré circuit ($\theta =5.{09}^{\circ }$) with 6×6 units, and the enlarge view is presented in the bottom chart. (b), (c) The measured impedances of a bulk node and an edge node in the frequency domain with $C_{\mathrm{inter}}/C_{\mathrm{intra}}=1$. Green regions mark nontrivial band gaps.

Figure 4

(a) The photograph image of the fabricated 1D moiré-ribbon circuit. (b), (c) The measured impedance spectra of a bulk node and a top edge node. (d), (e) The red and blue lines show the simulated and measured peak value and reaching time of the packet on different circuit nodes.

Figure 5

(a) The schematic diagram of the high-efficient transport of electronic signals assisted by topological edge states. (b), (c) The simulated and measured peak values of the packet arrived to edge and bulk nodes.