Landau quantization and Fermi velocity renormalization in twisted graphene bilayers

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Landau quantization and Fermi velocity renormalization in twisted graphene bilayers

Long-Jing Yin, Jia-Bin Qiao, Wen-Xiao Wang, Wei-Jie Zuo, Wei Yan, Rui Xu, Rui-Fen Dou, Jia-Cai Nie, and Lin He

Phys. Rev. B 92, 201408(R) – Published 10 November 2015

Abstract

Currently there is a lively discussion concerning Fermi velocity renormalization in twisted bilayers and several contradicted experimental results are reported. Here we study electronic structures of the twisted bilayers by scanning tunneling microscopy (STM) and spectroscopy (STS). The interlayer coupling strengths between the adjacent bilayers are measured according to energy separations of two pronounced low-energy van Hove singularities (VHSs) in the STS spectra. We demonstrate that there is a large range of values for the interlayer interaction not only in different twisted bilayers, but also in twisted bilayers with the same rotation angle. Below the VHSs, the observed Landau quantization in the twisted bilayers is identical to that of massless Dirac fermions in graphene monolayer, which allows us to measure the Fermi velocity directly. Our result indicates that the Fermi velocity of the twisted bilayers depends remarkably on both the twisted angles and the interlayer coupling strengths. This removes the discrepancy about the Fermi velocity renormalization in the twisted bilayers and provides a consistent interpretation of all current data.

Received 21 April 2015

DOI:https://doi.org/10.1103/PhysRevB.92.201408

Authors & Affiliations

Long-Jing Yin^1,2, Jia-Bin Qiao^1,2, Wen-Xiao Wang^1,2, Wei-Jie Zuo^1,2, Wei Yan^1,2, Rui Xu^1,2, Rui-Fen Dou¹, Jia-Cai Nie¹, and Lin He^1,2,*

¹Department of Physics, Beijing Normal University, Beijing, 100875, People's Republic of China
²Center for Advanced Quantum Studies, Beijing Normal University, Beijing, 100875, People's Republic of China

^*helin@bnu.edu.cn

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Vol. 92, Iss. 20 — 15 November 2015

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Images

Figure 1
(a)–(c) 20 nm × 20 nm STM topographic images of twisted bilayers with $θ = {(2.8 \pm 0.1)}^{\circ}, D = 5.0 \pm 0.2 nm$ (a); $θ = {(3.6 \pm 0.1)}^{\circ}, D = 3.92 \pm 0.08 nm$ (b); and $θ = {(6.0 \pm 0.1)}^{\circ}, D = 2.35 \pm 0.05 nm$ (c) on graphite surface. (d) Fast Fourier transforms image of (c). The outer and inner hexangular bright spots correspond to the reciprocal lattice of the graphene lattice and moiré patterns, respectively. The rotation angle $φ$ between the two reciprocal lattices is $27.0 \pm 0 . 1^{\circ}$ . The twisted angle can be estimated as $θ = (60^{\circ} - 2 φ) = {(6.0 \pm 0.2)}^{\circ}$ , which is consistent well with that measured in real space.
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Figure 2
(a) Several representative STS spectra of twisted graphene bilayers with various twisted angles. (b) STS spectra of the 3.6° twisted bilayer measured in different positions along the line in the moiré pattern (see the STM image in the inset). The values of VHSs are independent of the positions in the moiré pattern and similar results are observed in all of our samples. (c) Low-energy band structures of the 3.6° twisted bilayer with interlayer hopping $t_{θ} = 130$ and 50 meV. The inset shows the corresponding DOS of the twisted bilayer with different $t_{θ}$ . (d) Energy differences of the VHSs (the inset) and interlayer hopping as a function of the rotation angles in twisted graphene bilayers. The open symbols are data reported in literature. The circles represent data obtained on graphite surface and the squares represent data recorded on SiC surface.
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Figure 3
(a)–(c) STS spectra taken in various magnetic fields in twisted bilayers with $θ = {(2.8 \pm 0.1)}^{\circ}$ (a), $θ = {(3.6 \pm 0.1)}^{\circ}$ (b), and $θ = {(6.0 \pm 0.1)}^{\circ}$ (c). LL indices are marked. For clarity, all the spectra are offset on the $Y$ axis and all the curves are shifted to make the $n = 0 LLs$ stay at the same bias. (d)–(f) Landau level peak energies for different magnetic fields obtained in (a)–(c) plotted against $sgn (n) {(| n | B)}^{1 / 2}$ , as expected for massless Dirac fermions. The solid red lines are linear fitting of the data with the slopes yielding the Fermi velocity of $v_{F} = (1.03 \pm 0.02) \times 10^{6} m / s$ (d), $v_{F} = (0.811 \pm 0.004) \times 10^{6} m / s$ (e), and $v_{F} = (1.140 \pm 0.003) \times 10^{6} m / s$ (f), respectively.
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Figure 4
(a) Energy band dispersions for three twisted graphene bilayers with $θ = 6 . 0^{\circ}, θ = 3 . 9^{\circ}$ , and $θ = 2 . 9^{\circ}$ , respectively. Here $K$ is the Dirac point of the first layer and $K$ ' (or $K_{θ}$ ) is the Dirac point of the second layer. The separation of the two Dirac cones in reciprocal space is generated by the rotation angles between the graphene bilayers. The VHSs of the twisted bilayers appear at the $M$ point. In the calculation, the interlayer coupling strength for the three twisted bilayers is identical. The Fermi velocity is reduced with decreasing the twisted angles. (b) The Fermi velocity as a function of the twisted angles for different interlayer interactions. The solid circles and squares are the Fermi velocities of the twisted bilayers obtained on HOPG and SiC, respectively. The open symbols are the experimental data reported in literature, as illustrated in Fig. 2. The $v_{F} = 0$ for the 1.1° twisted bilayer was reported in Ref. [21]. The curves are the predicted Fermi velocity renormalization with different interlayer hopping strengths and we use different colors to denote different interlayer hopping strengths. The Fermi velocity is normalized with respect to $v_{F} = 1.10 \times 10^{6} m / s$ .
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