Quantum Hall Plateaus Research Articles

Observation of topological surface state quantum Hall effect in an intrinsic three-dimensional topological insulator

A three-dimensional (3D) topological insulator (TI) is a quantum state of matter with a gapped insulating bulk yet a conducting surface hosting topologically-protected gapless surface states. One of the most distinct electronic transport signatures predicted for such topological surface states (TSS) is a well-defined half-integer quantum Hall effect (QHE) in a magnetic field, where the surface Hall conductivities become quantized in units of (1/2)e2/h (e being the electron charge, h the Planck constant) concomitant with vanishing resistance. Here, we observe well-developed QHE arising from TSS in an intrinsic TI of BiSbTeSe2. Our samples exhibit surface dominated conduction even close to room temperature, while the bulk conduction is negligible. At low temperatures and high magnetic fields perpendicular to the top and bottom surfaces, we observe well-developed integer quantized Hall plateaus, where the two parallel surfaces each contributing a half integer e2/h quantized Hall (QH) conductance, accompanied by vanishing longitudinal resistance. When the bottom surface is gated to match the top surface in carrier density, only odd integer QH plateaus are observed, representing a half-integer QHE of two degenerate Dirac gases. This system provides an excellent platform to pursue a plethora of exotic physics and novel device applications predicted for TIs, ranging from magnetic monopoles and Majorana particles to dissipationless electronics and fault-tolerant quantum computers.

Current distribution and Hall potential landscape towards breakdown of the quantum Hall effect: a scanning force microscopy investigation

We present line- and area-scans of the Hall potential landscape of a two-dimensional electron system (2DES) in narrow (AlGa)As-based Hall bars under quantum Hall (QH) conditions, obtained by low-temperature scanning force microscopy. For several magnetic field values B in the regime of the QH plateau with Landau level filling factor , we measured the evolution of the Hall potential profiles and of the longitudinal voltage drop along the Hall bar as a function of increasing voltage/current bias, leading finally to the electrically induced breakdown of the quantum Hall effect (QHE). Basically two types of evolution were observed: for the low B-field side of the QHE plateau, two distinct Hall potential drops appear close to the two edges of a cross section, equally distributed at low bias but continuously developing to an asymmetrical distribution with increasing bias. At high bias, a steady increase of the longitudinal voltage drop is observed, accompanied by a rising slope of the Hall potential drop in the bulk. For the upper B-field side of the QH plateau, the Hall voltage drops are broadly distributed across the whole cross section, and the distribution remains almost unchanged until the bias reaches a critical value where the Hall potential profile changes rather abruptly, enhancing locally the Hall field. Beyond this, with further increase of the bias, a steep rise of the longitudinal voltage drop is detected. These findings are naturally explained in the microscopic picture of the QHE, based on the self-consistent evolution of the compressible and incompressible landscape inside the 2DES with increasing bias.

Experimental constraints and a possible quantum Hall state atν=5/2

Several topological orders have been proposed to explain the quantum Hall plateau at $\nu=5/2$. The observation of an upstream neutral mode on the sample edge [Bid et al., Nature (London) 466, 585 (2010)] supports the non-Abelian anti-Pfaffian state. On the other hand, the tunneling experiments [Radu et al., Science 320, 899 (2008); Lin et al., Phys. Rev. B 85, 165321 (2012); Baer et al., arXiv:1405.0428] favor the Halperin 331 state which exhibits no upstream modes. We find a topological order, compatible with the results of both types of experiments. That order allows both finite and zero spin polarizations. It is Abelian but its signatures in Aharonov-Bohm interferometry can be similar to those of the Pfaffian and anti-Pfaffian states.

Helical nuclear spin order in a strip of stripes in the quantum Hall regime

We investigate nuclear spin effects in a two-dimensional electron gas in the quantum Hall regime modeled by a weakly coupled array of interacting quantum wires. We show that the presence of hyperfine interaction between electron and nuclear spins in such wires can induce a phase transition, ordering electrons and nuclear spins into a helix in each wire. Electron-electron interaction effects, pronounced within the one-dimensional stripes, boost the transition temperature up to tens to hundreds of millikelvins in GaAs. We predict specific experimental signatures of the existence of nuclear spin order, for instance for the resistivity of the system at transitions between different quantum Hall plateaus.

Spin- and valley-dependent commensurability oscillations and electric-field-induced quantum Hall plateaux in periodically modulated silicene

We study the commensurability oscillations in silicene subject to a perpendicular electric field Ez, a weak magnetic field B, and a weak periodic potential V=V0cos(Cy),C=2π/a0 with a0 its period. The field Ez and/or the modulation lift the spin degeneracy of the Landau levels and lead to spin and valley resolved Weiss oscillations. The spin resolution is maximal when the field Ez is replaced by a periodic one Ez=E0cos(Dy),D=2π/b0, while the valley one is maximal for b0 = a0. In certain ranges of B values, the current is fully spin or valley polarized. Additional quantum Hall conductivity plateaux arise due to spin and valley intra-Landau-level transitions.

Pure scaling operators at the integer quantum Hall plateau transition.

Stationary wave functions at the transition between plateaus of the integer quantum Hall effect are known to exhibit multifractal statistics. Here we explore this critical behavior for the case of scattering states of the Chalker-Coddington network model with point contacts. We argue that moments formed from the wave amplitudes of critical scattering states decay as pure powers of the distance between the points of contact and observation. These moments in the continuum limit are proposed to be correlation functions of primary fields of an underlying conformal field theory. We check this proposal numerically by finite-size scaling. We also verify the conformal field theory prediction for a three-point function involving two primary fields.

Chalker scaling, level repulsion, and conformal invariance in critically delocalized quantum matter: Disordered topological superconductors and artificial graphene

We numerically investigate critically delocalized wavefunctions in models of 2D Dirac fermions, subject to vector potential disorder. These describe the surface states of 3D topological superconductors, and can also be realized through long-range correlated bond randomness in artificial materials like molecular graphene. A frozen regime can occur for strong disorder in these systems, wherein a single wavefunction presents a few localized peaks separated by macroscopic distances. Despite this rarefied spatial structure, we find robust correlations between eigenstates at different energies, at both weak and strong disorder. The associated level statistics are always approximately Wigner-Dyson. The system shows generalized Chalker (quantum critical) scaling, even when individual states are quasilocalized in space. We confirm analytical predictions for the density of states and multifractal spectra. For a single Dirac valley, we establish that finite energy states show universal multifractal spectra consistent with the integer quantum Hall plateau transition. A single Dirac fermion at finite energy can therefore behave as a Quantum Hall critical metal. For the case of two valleys and non-abelian disorder, we verify predictions of conformal field theory. Our results for the non-abelian case imply that both delocalization and conformal invariance are topologically-protected for multivalley topological superconductor surface states.

Scaling behavior of the insulator-to-plateau transition in a topological band model

The scaling behavior of the quantum phase transition from an insulator to a quantum Hall plateau state has often been examined within systems realizing Landau levels. We study the topological transition in energy band models with nonzero Chern number, which have the same topological property as a Landau level. We find that the topological band generally realizes the same universality class as the integer quantum Hall system in magnetic field for strong enough disorder scattering. Furthermore, the symmetry of the transition characterized by the relations: $\sigma_{xy}(E)=1-\sigma_{xy}(-E)$ for the Hall conductance and $\sigma_{xx}(E)=\sigma_{xx}(-E)$ for the longitudinal conductance is observed near the transition region. We also establish that the finite temperature dependence of the Hall conductance is determined by the inelastic scattering relaxation time, while the localization exponent $\nu$ remains unchanged by such scattering.

Statistics of conductances and subleading corrections to scaling near the integer quantum Hall plateau transition

We study the critical behavior near the integer quantum Hall plateau transition by focusing on the multifractal (MF) exponents Xq describing the scaling of the disorder-average moments of the point contact conductance T between two points of the sample, within the Chalker-Coddington network model. Past analytical work has related the exponents Xq to the MF exponents of the local density of states (LDOS). To verify this relation, we numerically determine the exponents Xq with high accuracy. We thereby provide, at the same time, independent numerical results for the MF exponents for the LDOS. The presence of subleading corrections to scaling makes such determination directly from scaling of the moments of T virtually impossible. We overcome this difficulty by using two recent advances. First, we construct pure scaling operators for the moments of T which have precisely the same leading scaling behavior, but no subleading contributions. Secondly, we take into account corrections to scaling from irrelevant (in the renormalization group sense) scaling fields by employing a numerical technique (“stability map”) recently developed by us. We thereby numerically confirm the relation between the two sets of exponents, Xq (point contact conductances) and (LDOS), and also determine the leading irrelevant (corrections to scaling) exponent y as well as other subleading exponents. Our results suggest a way to access multifractality in an experimental setting.

Phase Space for the Breakdown of the Quantum Hall Effect in Epitaxial Graphene

We report the phase space defined by the quantum Hall effect breakdown in polymer gated epitaxial graphene on SiC (SiC/G) as a function of temperature, current, carrier density, and magnetic fields up to 30 T. At 2 K, breakdown currents (I(c)) almost 2 orders of magnitude greater than in GaAs devices are observed. The phase boundary of the dissipationless state (ρ(xx)=0) shows a [1-(T/T(c))2] dependence and persists up to T(c)>45 K at 29 T. With magnetic field I(c) was found to increase ∝B(3/2) and T(c)∝B2. As the Fermi energy pproaches the Dirac point, the ν=2 quantized Hall plateau appears continuously from fields as low as 1 T up to at least 19 T due to a strong magnetic field dependence of the carrier density.

New Dirac points and multiple Landau level crossings in biased trilayer graphene

Recently a new high-mobility Dirac material, trilayer graphene, was realized experimentally. The band structure of $ABA$-stacked trilayer graphene consists of a monolayer-like and a bilayer-like pair of bands. Here we study electronic properties of $ABA$-stacked trilayer graphene biased by a perpendicular electric field. We find that the combination of the bias and trigonal warping gives rise to a set of new Dirac points: In each valley, seven species of Dirac fermions with small masses of order of a few meV emerge. The positions and masses of the emergent Dirac fermions are tunable by bias, and one group of Dirac fermions becomes massless at a certain bias value. Therefore, in contrast to bilayer graphene, the conductivity at the neutrality point is expected to show nonmonotonic behavior, becoming of the order of a few ${e}^{2}/h$ when some Dirac masses vanish. Further, we analyze the evolution of the Landau level spectrum as a function of bias. The emergence of new Dirac points in the band structure translates into new threefold-degenerate groups of Landau levels. This leads to an anomalous quantum Hall effect, in which some quantum Hall steps have a height of $3{e}^{2}/h$. At an intermediate bias, the degeneracies of all Landau levels get lifted, and in this regime all quantum Hall plateaus are spaced by ${e}^{2}/h$. Finally, we show that the pattern of Landau level crossings is very sensitive to certain band structure parameters, and can therefore provide a useful tool for determining their precise values.

Theory of inter-Landau-level magnetoexcitons in bilayer graphene

If bilayer graphene is placed in a strong perpendicular magnetic field, several quantum Hall plateaux are observed at low enough temperatures. Of these, the ${\ensuremath{\sigma}}_{xy}=4n{e}^{2}/h$ sequence ($n\ensuremath{\ne}0$) is explained by standard Landau quantization, while the other integer plateaux arise due to interactions. The low-energy excitations in both cases are magnetoexcitons, whose dispersion relation depends on single- and many-body effects in a complicated manner. Analyzing the magnetoexciton modes in bilayer graphene, we find that the mixing of different Landau level transitions not only renormalizes them, but essentially changes their spectra and orbital character at finite wavelength. These predictions can be probed in inelastic light scattering experiments.

Topological Blockade and Measurement of Topological Charge

The fractionally charged quasiparticles appearing in the 5/2 fractional quantum Hall plateau are predicted to have an extra nonlocal degree of freedom, known as topological charge. We show how this topological charge can block the tunneling of these particles, and how such topological blockade can be used to read out their topological charge. We argue that the short time scale required for this measurement is favorable for the detection of the non-Abelian anyonic statistics of the quasiparticles. We also show how topological blockade can be used to measure braiding statistics, and to couple a topological qubit with a conventional one.

Quantum phase transition between integer quantum Hall states of bosons

Bosonic integer quantum Hall (IQH) phases are a class of symmetry-protected topological (SPT) phases that, similar to the fermionic IQH states, support a quantized Hall conductance. They, however, require interactions for their realization. Here, we study quantum Hall plateau transitions between a trivial insulator and a bosonic IQH phase, in a clean system. Generically, we find an intervening superfluid phase. The presence of additional symmetries, however, can potentially lead to a direct transition between these phases. We employ a fermionic parton description that captures both the insulating phases as well as the transition between them. The critical theory is massless QED-3 with ${N}_{f}=2$ fermion flavors. The fermions have a surprisingly simple interpretation: they are vortices of the superfluid. Therefore, the universal conductivity at the transition, assuming it is continuous, equals the universal resistivity of Dirac fermions in QED-3. We also briefly comment on sigma model descriptions of the transition and the surface states of related three-dimensional SPT phases.

Integer quantum Hall effect on an interface with disclinations

The Hall conductivity of an electron gas on an interface showing topological defects as disclinations in the presence of an orthogonal constant magnetic field is investigated. This kind of defect induces either positive or negative singular curvature in the medium. It is shown that the positive curvature decreases the quantum Hall plateau widths and shifts the steps in the Hall conductivity to lower magnetic fields. In contrast, the negative one leaves to the existence of two types of plateaus, one with higher widths and the other one with lower widths in comparison to the flat case. In this case, the shift in the steps of the Hall conductivity goes to higher magnetic fields. We also investigate the Hall conductivity for electrons around a cylindrically symmetric distribution of disclinations and it turns out that it is the same as that corresponding to a single effective disclination.

Double trigonal warping and the anomalous quantum Hall step in bilayer graphene with Rashba spin–orbit coupling

We demonstrate that the trigonal warping observed in bilayer graphene is doubled in the presence of Rashba spin–orbit (RSO) coupling, i.e. the Dirac points along the three-fold symmetry axis are doubled. There are now seven Dirac points. Furthermore, the RSO interaction breaks the electron–hole symmetry of the magnetic band structure. The most intriguing feature is that the step of the quantum Hall plateau at zero energy is four times that at finite energy. The number of Dirac points and the zero energy Hall step are only determined by the existence of RSO coupling, but are independent of the strength of the coupling. The robustness of these phenomena suggests equivalence between the RSO coupling and the topological effect in bilayer coupling.

Unconventional Hall effect near charge neutrality point in a two-dimensional electron-hole system

The transport properties of the two-dimensional system in HgTe-based quantum wells containing simultaneously electrons and holes of low densities are examined. The Hall resistance, as a function of perpendicular magnetic field, reveals an unconventional behavior, different from the classical N-shaped dependence typical for bipolar systems with electron-hole asymmetry. The quantum features of magnetotransport are explained by means of numerical calculation of the Landau level spectrum based on the Kane Hamiltonian. The origin of the quantum Hall plateau {\sigma}xy = 0 near the charge neutrality point is attributed to special features of Landau quantization in our system.

Quantum Hall to Charge-Density-Wave Phase Transitions inABC-Trilayer Graphene

ABC-stacked trilayer graphene's chiral band structure results in three (n=0, 1, 2) Landau level orbitals with zero kinetic energy. This unique feature has important consequences on the interaction-driven states of the 12-fold degenerate (including spin and valley) N=0 Landau level. In particular, at many filling factors ν(T) = ±5, ±4, ±2, ±1 a quantum phase transition from a quantum Hall liquid state to a triangular charge-density wave occurs as a function of the single particle-induced Landau level orbital splitting Δ(LL). Experimental signatures of this phase transition are also discussed.

Edge state transport through disordered graphene nanoribbons in the quantum Hall regime

The presence of strong disorder in graphene nanoribbons yields low-mobility diffusive transport at high charge densities, whereas a transport gap occurs at low densities. Here, we investigate the longitudinal and transverse magnetoresistance of a narrow ($\ensuremath{\sim}$60 nm) nanoribbon in a six-terminal Hall bar geometry. At $B=$ 11 T, quantum Hall plateaus appear at ${\ensuremath{\sigma}}_{xy}=\ifmmode\pm\else\textpm\fi{}2{e}^{2}/h$, $\ifmmode\pm\else\textpm\fi{}6{e}^{2}/h$, and $\ifmmode\pm\else\textpm\fi{}10{e}^{2}/h$, for which the Landau-level spacing is larger than the Landau-level broadening. Interestingly, the transport gap does not disappear in the quantum Hall regime, when the zero-energy Landau level is present at the charge neutrality point, implying that it cannot originate from a lateral confinement gap. At high charge densities, the longitudinal and Hall resistance exhibit reproducible fluctuations, which are most pronounced at the transition regions between Hall plateaus. Bias-dependent measurements strongly indicate that these fluctuations can be attributed to phase-coherent scattering in the disordered ribbon.

Thermopower of quantum Hall states in Corbino geometry as a measure of quasiparticle entropy

Using the Onsager relation between electric and heat transport coefficients, and considering the very different roles played by the quantum Hall condensate and quasiparticles in transport, we argue that near the center of a quantum Hall plateau thermopower in a Corbino geometry measures entropy per quasiparticle per quasiparticle charge. This relation indicates that thermopower measurement in a Corbino setup is a more direct measure of quasiparticle entropy than in a Hall bar. Treating disorder within the self-consistent Born approximation, we show through an explicit microscopic calculation that this relation holds on an integer quantum Hall plateau at low temperatures. Applying this to non-Abelian quantum Hall states, we argue that Corbino thermopower at sufficiently low temperature becomes temperature independent and measures the quantum dimension of non-Abelian quasiparticles that determines the topological entropy they carry.