Milli-Tesla quantization enabled by tuneable Coulomb screening in large-angle twisted graphene

Abstract

The electronic quality of graphene has improved significantly over the past two decades, revealing novel phenomena. However, even state-of-the-art devices exhibit substantial spatial charge fluctuations originating from charged defects inside the encapsulating crystals, limiting their performance. Here, we overcome this issue by assembling devices in which graphene is encapsulated by other graphene layers while remaining electronically decoupled from them via a large twist angle (~10-30°). Doping of the encapsulating graphene layer introduces strong Coulomb screening, maximized by the sub-nanometer distance between the layers, and reduces the inhomogeneity in the adjacent layer to just a few carriers per square micrometre. The enhanced quality manifests in Landau quantization emerging at magnetic fields as low as ~5 milli-Tesla and enables resolution of a small energy gap at the Dirac point. Our encapsulation approach can be extended to other two-dimensional systems, enabling further exploration of the electronic properties of ultrapure devices.

Fig. 1. Characterization of a large-angle twisted bilayer graphene (LATBG) device.

a Top panel: schematic structure of LATBG device, utilizing thin hBN flakes as dielectrics for bottom graphite and top metal gates, $V_{\text{tg}}$ and $V_{\text{bg}}$ are top and bottom gate voltages; bottom panel: optical image of LATBG Hall bar device. Black line highlights the metallic top gate and electrical contacts. Scale bar is 1 μm. b Longitudinal resistivity as a function of total charge density $n_{\text{tot}}$ for D = 0 V/nm, and D = 0.5 V/nm measured at zero magnetic field. Dashed circles indicate the positions of charge neutrality points of top (t-CNP, black circle) and bottom graphene layers (b-CNP, blue circle) calculated using the electrostatic model (see Supplementary Note 1). Insets schematically show band structure of LATBG under applied D when the Fermi level crosses the CNP of one of the graphene layers; here, the red part of the cone represents hole doping, and blue represents electron doping of graphene. c Longitudinal resistivity at B = 0 T as a function of $n_{\text{tot}}$ and D. Dashed lines indicate expected positions of CNPs. d, e Resistivity as a function of magnetic field B and $n_{\text{tot}}$ for D = 0 V/nm (d) and D = 0.6 V/nm (e). In (d) dashed lines show the expected position of doubled graphene filling factors. In (e) white dashed lines are a guide for the eye of the first three Landau levels and CNP gap boundaries in bottom graphene layer. See the text for the further discussion. Further examples of Landau fan measurements are shown in Supplementary Figs. 2 and 4. The high magnetoresistance observed around $n_{\text{tot}}=0$ can be attributed to the compensated semimetal state, similar to ref. , which naturally forms under applied D when one layer is electron-doped and the other is hole-doped. Measurements were performed at 2 K for all panels.

Fig. 2. Resolving the onset of Landau quantization in milli-Tesla magnetic field.

a Schematic illustration of LATBG band structure at low magnetic field B and high D when Fermi level is tuned close to the CNP of one of graphene layers. b Fan diagram measured at D = −0.75 V/nm and 2 K shown as a function of chemical potential in the bottom graphene layer ${\mu }_{\text{b}}$. Black parabolic dashed lines indicate the expected position for the first five Landau levels (LLs) plotted using a standard graphene sequence $E_{\text{N}}=\sqrt{2\hslash eB{v}_{\text{F}}^{2}N+{\Delta }^2}$, where $\hslash$ is the reduced Plank constant, $e$ is the electron charge, $E_{\text{N}}$ is the energy of the $N^{\text{th}}$ LL, and $v_{\text{F}}$ is the Fermi velocity in graphene, $2\Delta$ is a band gap discussed further in the text. The horizontal dashed line and the error bar mark the onset of Landau quantisation. Square root dashed lines indicate the limit of quantization set by the probe width (see Supplementary Note 5). c Schematic illustrations of electron-hole puddles in LATBG at zero (top) and under applied (bottom) displacement fields. Red and blue shaded regions represent positive and negative doping correspondingly, coloured circles in the hBN illustrate charged defects. d Simulated charge density profiles in top graphene calculated for hBN with impurity density $n_{\text{imp}}={10}^{10}{\text{cm}}^{-2}$ at zero doping of the bottom layer n_b = 0, and for n_b = 0.5 × 10¹² cm⁻² (see Supplementary Note 6). Scale bars are 250 nm. e Modelling of LATBG resistance subjected to high D under the assumption of a gapped graphene spectra (see Supplementary Note 7).

Fig. 3. High quality of LATTG devices.

a Schematic structure of LATTG devices, graphene layers (b, m, and t labels indicate bottom, middle, and top layers, respectively) are twisted by large angles ${\theta }_{12}$ (the angle between the bottom and the middle layers) and ${\theta }_{23}$ (the angle between the middle and the top layers). See “Methods” for further information. Optical images show two measured LATTG Hall bar devices A and B, scale bar is 5 μm. b Longitudinal resistance at B = 0 T as a function of charge density $n_{\text{tot}}$ and displacement field D measured in device A. Coloured lines indicate conditions of CNP for top, middle, and bottom layers with black, red, and yellow correspondingly. Inset band structures illustrate the position of CNPs of each layer. c Longitudinal resistance at B = 50 mT as a function of $n_{\text{tot}}$ and D measured in device A. Coloured dashed lines show the calculated CNP positions for all three layers based on the electrostatic model described in the Supplementary Note 1. We therefore label the nearest Landau levels as ±1, ±2, with the sign reflecting the charge carrier type in each layer. d–g Magnetoresistance measurements for devices A and B. d, f Measured for fixed D = −0.5 V/nm and D = 0.5 V/nm correspondingly as a function of $n_{\text{tot}}$, coloured arrows show positions of CNPs. e, g Longitudinal resistance measured for fixed $n_{\text{tot}}$ as a function of D with removed background (see Supplementary Fig. 10 for more details). Onset of Landau quantisation become resolvable already at $B^{*}=5-7$ mT in (e) and at $B^{*}=5-6$ mT in (g). Measurements were done at 2 K for all panels.