Terahertz conductivity of graphene on boron nitride

Ashley M. DaSilva, Jeil Jung, Shaffique Adam, and Allan H. MacDonald

Phys. Rev. B 92, 155406 – Published 5 October 2015

Abstract

The conductivity of graphene on a boron nitride substrate exhibits features in the terahertz (THz) and infrared (IR) frequency regimes that are associated with the periodic moiré pattern formed by weakly coupled two-dimensional materials. The THz and IR features are strongest when the two honeycomb lattices are orientationally aligned, and in this case they are Pauli blocked unless the Fermi level is close to $\pm 150$ meV relative to the graphene sheet Dirac point. Because the transition energies between moiré bands formed above the Dirac point are small, ac conductivity features in $n$ -doped graphene tend to be overwhelmed by the Drude peak. The substrate-induced band splitting is larger at energies below the Dirac point, however, and it can lead to sharp features at THz and IR frequencies in $p$ -doped graphene. In this paper, we focus on the strongest few THz and IR features, explaining how they arise from critical points in the moiré-band joint density of states, and commenting on the interval of Fermi energy over which they are active.

Received 12 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.155406

Authors & Affiliations

Ashley M. DaSilva¹, Jeil Jung², Shaffique Adam^3,4, and Allan H. MacDonald¹

¹Department of Physics, The University of Texas at Austin, Austin, Texas 78712-1192, USA
²Department of Physics, University of Seoul, Seoul 130-743, Korea
³Centre for Advanced 2D Materials and Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
⁴Yale-NUS College, 16 College Avenue West, Singapore 138527

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Vol. 92, Iss. 15 — 15 October 2015

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Images

Figure 1
Pauli blocking in bare graphene. (a) This illustration shows the Dirac cone of doped isolated graphene with a Fermi level $ɛ_{F}$ given by the dashed line. Blue arrows represent interband transitions from the valence band to the conduction band. The middle arrow, with frequency $2 ω_{F} = 2 ɛ_{F} / ℏ$ , is at the minimum frequency for interband transitions. Below $2 ω_{F}$ , transitions are not allowed due to the Pauli exclusion principle. (b) Density of states of aligned graphene on hBN. The shaded regions represent the locations of the substrate-induced gapped Dirac cone replicas. The conductivity is strongly Fermi-level-dependent near these energies because of Pauli blocking.
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Figure 2
Band structure and ac conductivity of graphene on boron nitride. (a) Band structure of orientationally aligned graphene on boron nitride. The dashed lines show the Fermi levels for which the ac conductivity has been calculated. (b) Schematic picture of the moiré Brillouin zone, outlined in red. The bands are plotted along the black lines in momentum space. Blue dots represent the first shell of moiré reciprocal-lattice vectors. (c) The ac conductivity when the Fermi level is at the Dirac point $ɛ_{F} = 0$ . The red dashed line shows that $σ (ω)$ approaches the bare graphene value $σ_{0} = π e^{2} / 2 h$ at high frequencies. (d) The ac conductivity when the Fermi level is in the conduction band and near the secondary Dirac point with $ɛ_{F} = ℏ v G / \sqrt{3}$ (black) and near the secondary Dirac point with $ɛ_{F} = ℏ v G / 2$ (blue). $G$ is the magnitude of the moiré reciprocal-lattice vectors in the first shell. (e) The ac conductivity when the Fermi level is in the valence band and near the secondary Dirac point with $ɛ_{F} = - ℏ v G / \sqrt{3}$ (black) and $ɛ_{F} = - ℏ v G / 2$ (blue).
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Figure 3
Valence-band conductivity features association with transitions at high-symmetry points in the moiré Brillouin zone. The interband transition energy (first row), band structure (second row), and conductivity (third and fourth rows) for interband transitions between three pairs of valence bands. The left column corresponds to transitions from the second to the first valence band, the middle column corresponds to transitions from the third to the first valence band, and the right column corresponds to transitions from the third to the second valence band. In the band structures (second row), dominant transitions have been identified with colored arrows, and their corresponding energies are marked as vertical dashed lines on the conductivity plots (third and fourth rows). The two Fermi energies at which calculations were performed, marked by black horizontal dashed lines on the band-structure plots, are $ɛ_{1} = - ℏ v G / \sqrt{3}$ and $ɛ_{2} = - ℏ v G / 2$ . The third row corresponds to Fermi energy $ɛ_{F} = ɛ_{1}$ , while the fourth row corresponds to Fermi energy $ɛ_{F} = ɛ_{2}$ .
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