3D to 2D localization in supertwisted multilayers

Abstract

We study the electronic properties of multilayer "spirals" of two-dimensional materials with continuously increasing twist angle. The electronic states are shown to undergo a universal 3D-to-2D transition on increasing the in-plane momentum ${\bf k}_\parallel$ away from the $Γ$ point. The states with $k_\parallel>k_c$ are localized along the z axis due to mismatch between electronic dispersions of the twisted layers, whereas those with $k_\parallel<k_c$ are extended. We support our results by mapping of the system on the Aubry-André model and deduce the experimental signatures of 3D to 2D localization in transport experiments.

Figures (9)

• Figure 1: 3D supertwisted spirals localization. (a) The schematics of the layer stacking of a crystal with $\theta$ twist between layers. The ellipses depict the electron Fermi surfaces for each layer. For each in-plane momentum $\vb{k} = (k_x, k_y)$, the system can be modeled as a tight-binding model with varying on-site potentials. (b) IPR as a function of $k_x$ and $k_y$ at $L = 20$, $\theta = 0.21\pi \approx 39^\circ$, and $\varepsilon = 0.1$. (c) average IPR as a function of $k$ at $L = 20$, $\theta = 0.21\pi$, and $\varepsilon = 0.1$.
• Figure 2: Spectra and conductance. (a) The energy spectra of the system at $\theta = (1 + \sqrt{5})\pi/15 \approx 0.21 \pi \approx 39^\circ$. The color indicate IPR of the state. (b) The normalized conductance $G/G_0$ as a function of doping $E_F$ at $t =0.03$, $L = 20$, and $\varepsilon = 0.1$. $G_0$ is the ideal ballistic conductance $G_0 = (e^2/h) N_e$, where $N_e = k_{F,lead}^2/(4\pi)$ is the number of states supported by the lead. Both subplots are labeled by the energy region into I. subspectra regime, II. doping regime, III. localization regime, and IV. universal decay regime.
• Figure 3: Universal conductance decays. (a) Normalized conductance $G/G_0$ as a function of $E_F$ at different $(L, \theta, \varepsilon, t)$. $G_0$ is the ideal ballistic conductance. The $\theta$ are chosen such that $\beta$ is an irrational number $\theta = (1+\sqrt{5})/15 \pi \approx 0.21 \pi \approx 39^\circ$, $\theta = (1+\sqrt{5})/30 \pi \approx 0.10 \pi \approx 18^\circ$, and $\theta = (1+\sqrt{5})/30 \pi \approx 0.06 \pi \approx 11^\circ$. (b) logarithm of rescaled conductance $\log G/t$ as a function of $L \log (E_F/E_c)$ at different $(L, \theta, \varepsilon, t)$ (same legend with (a)). The dashed line is the predicted empirical relation $G = 0.8 (e^2/h) \mathcal{N}t (E_c/E_F)^L$.
• Figure 4: Conductance as a function of $\beta$. (a) $G/G_0$ as a function of $E_F$ at a series of $\theta$ which are rational approximant of $\theta = (1+\sqrt{5})/15 \pi$. (b) Normalized conductance $G/G_0$ as a function of $\theta$ at different $L$. The calculation is performed at $t = 0.03$, $\varepsilon = 0.1$, and $E_F = 1.0$. Notable peaks are labeled by $\beta$ as simple fractions.
• Figure 5: Localization in Aubry-André model at small $\beta$. (a) Schematics of the eigenstates of AA model in the localized phase $\Delta = 2.5 t$. (top) the layer-dependent potential $\epsilon_l$ as a function of layer $l$. (bottom) The wavefunction eigenstates $|\psi_l|$ as a function of layer number $l$ (shifted for clarity). We can see that the eigenstates forms bound states near the minima of $\epsilon_l$ which locally resembled a harmonic oscillator in a small $\beta$ limit. (b) The decay of the wavefunction $|\psi_l|$ at spectra minima when $\Delta = 2.5 t$ and $\beta = (1 + \sqrt{5})/100$. Here, the local form of the wavefunction shows $\beta, \Delta$-dependent feature, while the long-range features show exponential decays defines from $\langle \psi_{j} \rangle \sim \exp(- l / \xi)$ (dashed line) where $1/\xi = \log (2\Delta/t)$.