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A twist between superposed graphene layers gives rise to two peaks in the density of states (van Hove singularities) which flank the charge neutrality point and are separated from each other by an energy which increases with twist angle [13]. For twist angles exceeding 10, the low energy spectrum (< 1 {\rm} e {\rm} V) is indistinguishable from that of single layer graphene. The absence of van Hove singularities and the single layer LL spectrum in the data reported here provide direct evidence of layer decoupling. Although there is no topographic signature of the associated Moiré pattern, which would require a very sharp tip, the above signatures are taken as evidence for a large twist angle.
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A twist between superposed graphene layers gives rise to two peaks in the density of states (van Hove singularities) which flank the charge neutrality point and are separated from each other by an energy which increases with twist angle [13]. For twist angles exceeding 10, the low energy spectrum (< 1 {\rm} e {\rm} V) is indistinguishable from that of single layer graphene. The absence of van Hove singularities and the single layer LL spectrum in the data reported here provide direct evidence of layer decoupling. Although there is no topographic signature of the associated Moiré pattern, which would require a very sharp tip, the above signatures are taken as evidence for a large twist angle.
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Although STM explores only a small area of the sample, the gate-voltage dependence of the data in Fig. 3(a) reflects the available states in the entire sample including those that are outside the field of view of the STM. This is because the gate covers the entire sample and can populate all available states.
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Although STM explores only a small area of the sample, the gate-voltage dependence of the data in Fig. 3(a) reflects the available states in the entire sample including those that are outside the field of view of the STM. This is because the gate covers the entire sample and can populate all available states.
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These properties are readily understood by examining the local DOS, D s (E, r) = ∫ d 2 r D (E, r) / S, averaged over a finite-size region S around the impurity. Unlike the DOS averaged over the whole sample, D s (E, r), is manifestly particle-hole asymmetric within a given LL, which translates to the particle-hole asymmetry of the local screening.
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These properties are readily understood by examining the local DOS, D s (E, r) = ∫ d 2 r D (E, r) / S, averaged over a finite-size region S around the impurity. Unlike the DOS averaged over the whole sample, D s (E, r), is manifestly particle-hole asymmetric within a given LL, which translates to the particle-hole asymmetry of the local screening.
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D (E, r) = 4 Σ N m δ γ (E - E N m) ψ N m † (r) ψ N m (r). Here, δ γ (E - E N m) = γ / {π [(E - E N m) 2 + γ 2]} represents the broadened LL. The peak intensity is determined by the probability density ψ N m † (r) ψ N m (r) and is position dependent.
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E 00 = (Z / κ) (e 2 / 4 π ε 0 l B) (π / 2) (1 / 2) [1 - E r f (a / l B)], where E r f (x) is the error function.
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Δ E 00 = (Z / κ) (e 2 / 4 π ε 0 l B) (π / 2) (1 / 2) [1-E r f (a / l B)], where E r f (x) is the error function.
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This value is comparable to the zero-field RPA estimate by the authors of Ref. [26] for double layer graphene, κ g r = 1 + g l g s g v π r s / 8 ≈ 3.75, suggesting that when the LLs are almost empty, screening of positive charges in graphene is comparable to the zero-field case. Here, r s = 4 π e 2 / h v F (κ S i O 2 + 1) is the dimensionless Wigner-Seitz radius which measures the relative strength of the potential and kinetic energies in an interacting quantum Coulomb system with linear dispersion. We note that for single layer graphene, g l = 1, screening would be significantly weaker, κ g r ≈ 2.4.
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This value is comparable to the zero-field RPA estimate by the authors of Ref. [26] for double layer graphene, κ g r = 1 + g l g s g v π r s / 8 ≈ 3.75, suggesting that when the LLs are almost empty, screening of positive charges in graphene is comparable to the zero-field case. Here, r s = 4 π e 2 / h v F (κ S i O 2 + 1) is the dimensionless Wigner-Seitz radius which measures the relative strength of the potential and kinetic energies in an interacting quantum Coulomb system with linear dispersion. We note that for single layer graphene, g l = 1, screening would be significantly weaker, κ g r ≈ 2.4.
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