Integer Quantum Hall Effect in Trilayer Graphene (original) (raw)

Experimental observation of the quantum Hall effect and Berry's phase in graphene

Nature, 2005

When electrons are confined in two-dimensional materials, quantum-mechanically enhanced transport phenomena such as the quantum Hall effect can be observed. Graphene, consisting of an isolated single atomic layer of graphite, is an ideal realization of such a two-dimensional system. However, its behaviour is expected to differ markedly from the well-studied case of quantum wells in conventional semiconductor interfaces. This difference arises from the unique electronic properties of graphene, which exhibits electron-hole degeneracy and vanishing carrier mass near the point of charge neutrality 1,2 . Indeed, a distinctive half-integer quantum Hall effect has been predicted 3-5 theoretically, as has the existence of a non-zero Berry's phase (a geometric quantum phase) of the electron wavefunction-a consequence of the exceptional topology of the graphene band structure 6,7 . Recent advances in micromechanical extraction and fabrication techniques for graphite structures 8-12 now permit such exotic two-dimensional electron systems to be probed experimentally. Here we report an experimental investigation of magneto-transport in a high-mobility single layer of graphene. Adjusting the chemical potential with the use of the electric field effect, we observe an unusual halfinteger quantum Hall effect for both electron and hole carriers in graphene. The relevance of Berry's phase to these experiments is confirmed by magneto-oscillations. In addition to their purely scientific interest, these unusual quantum transport phenomena may lead to new applications in carbon-based electronic and magneto-electronic devices.

Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene

Nature Physics, 2006

T here are two known distinct types of the integer quantum Hall effect. One is the conventional quantum Hall effect, characteristic of two-dimensional semiconductor systems 1,2 , and the other is its relativistic counterpart observed in graphene, where charge carriers mimic Dirac fermions characterized by Berry's phase π, which results in shifted positions of the Hall plateaus 3-9 . Here we report a third type of the integer quantum Hall effect. Charge carriers in bilayer graphene have a parabolic energy spectrum but are chiral and show Berry's phase 2π affecting their quantum dynamics. The Landau quantization of these fermions results in plateaus in Hall conductivity at standard integer positions, but the last (zero-level) plateau is missing. The zero-level anomaly is accompanied by metallic conductivity in the limit of low concentrations and high magnetic fields, in stark contrast to the conventional, insulating behaviour in this regime. The revealed chiral fermions have no known analogues and present an intriguing case for quantum-mechanical studies. provides a schematic overview of the quantum Hall effect (QHE) behaviour observed in bilayer graphene by comparing it with the conventional integer QHE. In the standard theory, each filled single-degenerate Landau level contributes one conductance quantum e 2 /h towards the observable Hall conductivity (here e is the electron charge and h is Planck's constant). The conventional QHE is shown in , where plateaus in Hall conductivity σ xy make up an uninterrupted ladder of equidistant steps. In bilayer graphene, QHE plateaus follow the same ladder but the plateau at zero σ xy is markedly absent . Instead, the Hall conductivity undergoes a double-sized step across this region. In addition, longitudinal conductivity σ xx in bilayer graphene remains of the order of e 2 /h, even at zero σ xy . The origin of the unconventional QHE behaviour lies in the coupling between two graphene layers, which transforms massless Dirac fermions, characteristic of single-layer graphene 3-9 , into a new type of chiral quasiparticle. Such quasiparticles have an ordinary parabolic spectrum ε(p) = p 2 /2m with effective mass m, but

Half integer quantum Hall effect in high mobility single layer epitaxial graphene

Applied Physics Letters, 2009

The quantum Hall effect, with a Berry's phase of π is demonstrated here on a single graphene layer grown on the C-face of 4H silicon carbide. The mobility is ∼ 20,000 cm 2 /V·s at 4 K and 15,000 cm 2 /V·s at 300 K despite contamination and substrate steps. This is comparable to the best exfoliated graphene flakes on SiO 2 and an order of magnitude larger than Si-face epitaxial graphene monolayers. These and other

Integer quantum Hall effect in gapped single-layer graphene

Physical Review B, 2012

Analytical expressions for the Hall conductivity σ yx and the longitudinal resistivity ρ xx are derived in gapped, single-layer graphene using linear response theory. The gap 2 , described by a mass term, is induced by a substrate made of hexagonal boron nitride (h-BN) and produces two levels at ±. It is shown that σ yx has the same form as for a graphene sample supported by a common substrate without a mass term. The differences are a shift in the energy spectrum, which is not symmetric with respect to the Dirac point for either valley due to the gap, the absence of a zero-energy Landau level, and the nonequivalence of the K and K valleys. In addition, the dispersion of the energy levels, caused by electron scattering by impurities, modifies mostly plateaus due to the levels at ±. It is shown that the resistivity ρ xx exhibits an oscillatory dependence on the electron concentration. The main difference with the usual graphene samples, on SiO 2 substrates, occurs near zero concentration, as the energy spectra differ mostly near the Dirac point.

Insulator-quantum Hall transitionin monolayer epitaxial graphene

RSC advances, 2016

We report on magneto-transport measurements on low-density, large-area monolayer epitaxial graphene devices grown on SiC. We observe temperature (T)-independent crossing points in the longitudinal resistivity ρxx, which are signatures of the insulator-quantum Hall (I-QH) transition, in all three devices. Upon converting the raw data into longitudinal and Hall conductivities σxx and σxy, in the most disordered device, we observed T-driven flow diagram approximated by the semi-circle law as well as the T-independent point in σxy near e(2)/h. We discuss our experimental results in the context of the evolution of the zero-energy Landau level at low magnetic fields B. We also compare the observed strongly insulating behaviour with metallic behaviour and the absence of the I-QH transition in graphene on SiO2 prepared by mechanical exfoliation.

Quantum anomalous Hall effect in single-layer and bilayer graphene

Physical Review B, 2011

The quantum anomalous Hall effect can occur in single and few layer graphene systems that have both exchange fields and spin-orbit coupling. In this paper, we present a study of the quantum anomalous Hall effect in single-layer and gated bilayer graphene systems with Rashba spin-orbit coupling. We compute Berry curvatures at each valley point and find that for single-layer graphene the Hall conductivity is quantized at σxy = 2e 2 /h, with each valley contributing a unit conductance and a corresponding chiral edge state. In bilayer graphene, we find that the quantized anomalous Hall conductivity is twice that of the single-layer case when the gate voltage U is smaller than the exchange field M , and zero otherwise. Although the Chern number vanishes when U > M , the system still exhibits a quantized valley Hall effect, with the edge states in opposite valleys propagating in opposite directions. The possibility of tuning between different topological states with an external gate voltage suggests possible graphene-based spintronics applications.

Landau-level dispersion and the quantum Hall plateaus in bilayer graphene

2013

We study the quantum Hall effect (QHE) in bilayer graphene using the Kubo-Greenwood formula. At zero temperature the Hall conductivity σ yx is given by σ yx = 4(N + 1)e 2 /h with N the index of the highest occupied Landau level (LL). Including the dispersion of the LLs and their width, due to e.g. scattering by impurities, produces the plateau of the n = 0 LL in agreement with experimental results on doped samples and similar theoretical results on single-layer graphene plateaus widen with impurity concentration. Further, the evaluated resistivity ρ xx exhibits a strong, oscillatory dependence on the electron concentration. Explicit results are obtained for δ -function impurities.

Bilayer-induced asymmetric quantum Hall effect in epitaxial graphene

The transport properties of epitaxial graphene on SiC(0001) at quantizing magnetic fields are investigated. Devices patterned perpendicularly to SiC terraces clearly exhibit bilayer inclusions distributed along the substrate step edges. We show that the transport properties in the quantum Hall regime are heavily affected by the presence of bilayer inclusions, and observe a significant departure from the conventional quantum Hall characteristics. A quantitative model involving enhanced inter-channel scattering mediated by the presence of bilayer inclusions is presented that successfully explains the observed symmetry properties.

Anomalous integer quantum Hall effect in AA-stacked bilayer graphene

Physical Review B, 2010

Recent experiments indicate that AA-stacked bilayer graphenes (BLG) could exist. Since the energy bands of the AA-stacked BLG are different from both the monolayer and AB-stacked bilayer graphenes, different integer quantum Hall effect in the AA-stacked graphene is expected. We have therefore calculated the quantized Hall conductivity σxy and also longitudinal conductivity σxx of the AA-stacked BLG within the linear response Kubo formalism. Interestingly, we find that the AAstacked BLG could exhibit both conventional insulating behavior (theν = 0 plateau) and chirality for |μ| < t, whereν is the filling factor (ν = σxyh/e 2),μ is the chemical potential, and t is the interlayer hopping energy, in striking contrast to the monlayer graphene (MLG) and AB-stacked BLG. We also find that for |μ| = [(√ n2 + √ n1)/(√ n2 − √ n1)]t, where n1 = 1, 2, 3, • • •, n2 = 2, 3, 4, • • • and n2 > n1, the Hall conductivity is quantized as σxy = ± 4e 2 h n, n = 0, 1, 2, • • •, if |μ| < t and σxy = ± 4e 2 h n, n = 1, 2, 3, • • •, if |μ| > t. However, if |μ| = [(√ n1 + √ n2)/(√ n2 − √ n1)]t, theν = ±4(n1 + n2)n plateaus are absent, where n = 1, 2, 3, • • •, in comparison with the ABstacked BLG within the two-band approximation. We show that in the low-disorder and highmagnetic-field regime, σxx → 0 as long as the Fermi level is not close to a Dirac point, where Γ denotes the Landau level broadening induced by disorder. Furthermore, when σxy is plotted as a function ofμ, aν = 0 plateau appears acrossμ = 0 and it would disappear if the magnetic field B = πt 2 /N ehυ 2 F , N = 1, 2, 3, • • •. Finally, the disappearance of the zero-Hall conductivity plateau is always accompanied by the occurence of a 8e 2 /h-step atμ = t.