Incompressible Strips Research Articles

Revision of the edge channel picture for the integer quantum Hall effect

State of the art computing opens now a new window to the integer quantum Hall effect (IQHE) regime, which enforces a major revision of the common knowledge accumulated so far. In our record-breaking application of the Hartree–Fock method we use up to 3000 electrons distributed over up to 5000 states for almost macroscopic system size of 1000 × 1000 nm. In particular, the formation of compressible and in-compressible edge stripes turns out to develop essentially different from the common picture used so far. Oppositely to the theory of Chklovskii, Shklovskii and Glazman (CSG), the narrow channels, as assumed by the early models of the IQHE, do not widen up into wide compressible stripes. Instead, the wide compressible stripes of CSG transform into a mixture of clusters of full and empty spin-split LLs, while the cluster boundaries create a network of still narrow quantum channels sitting on top of the wide compressible stripes. On this background the early models based on narrow edge channels do not suffer from neglecting electron–electron interaction as falsely stated in the past. Quite oppositely, in contrast to the common believe, our modeling demonstrates that also the IQHE regime carries the hallmark of many-body physics which stabilizes narrow edge channels also in the presence of electron–electron interaction.

Edge Reconstruction of a Time-Reversal Invariant Insulator: Compressible-Incompressible Stripes.

Two-dimensional (2D) topological electronic insulators are known to give rise to gapless edge modes, which underlie low energy dynamics, including electrical and thermal transport. This has been thoroughly investigated in the context of quantum Hall phases, and time-reversal invariant topological insulators. Here we study the edge of a 2D, topologically trivial insulating phase, as a function of the strength of the electronic interactions and the steepness of the confining potential. For sufficiently smooth confining potentials, alternating compressible and incompressible stripes appear at the edge. Our findings signal the emergence of gapless edge modes which may give rise to finite conductance at the edge. This would suggest a novel scenario of a nontopological metal-insulator transition in clean 2D systems. The incompressible stripes appear at commensurate fillings and may exhibit broken translational invariance along the edge in the form of charge density wave ordering. These are separated by structureless compressible stripes.

Imaging disorder-induced scattering centers in quantum Hall incompressible strip

While the disorder-induced quantum Hall (QH) effect has been studied previously, the effect ofdisorder potential on microscopic features of the integer QH effect remains unclear, particularly forthe incompressible (IC) strip. In this research, a scanning gate microscope incorporated with thenonequilibrium transport technique is used to image the region of QH IC strip that emerges near thesample edge. It was found that different mobility samples with varying disorder potentials showedthe same spatial dependence of the IC strip on the filling factor ({\nu}). In the low-mobility samplealone, scattering centers such, bright, dark and annular patterns, alternately appear within the IC strip. These observed patterns are ascribed to inter-LL scattering assisted by resonance tunnelingthrough an impurity bound state. It is concluded that disorder-induced scattering can be effectivelydetected using the applied technique in a low-mobility sample.

The effect of disorder on local electron temperature in quantum Hall systems

The local electron temperature distribution is calculated considering a two dimensional electron system in the integer quantum Hall regime in presence of disorder and uniform perpendicular magnetic fields. We solve thermo-hydrodynamic equations to obtain the spatial distribution of the local electron temperature in the linear-response regime. It is observed that, the variations of electron temperature exhibit an antisymmetry regarding the center of the sample in accordance with the location of incompressible strips. To understand the effect of sample mobility on the local electron temperature we impose a disorder potential calculated within the screening theory. Here, long range potential fluctuations are assumed to simulate cumulative disorder potential depending on the impurity atoms. We observe that the local electron temperature strongly depends on the number of impurities in narrow samples. The electron temperature $$T_e$$ versus position calculated for different values of the modulation potential $$V_0$$ . The calculations are done at $$T_\mathrm{L} = 0.04 \,E_F^0/k_B$$ lattice temperature considering impurity $$N_l = 6600$$ and repeated for three different values of magnetic field $$\hbar \omega _c/E_F^0 = 0.80$$ , 0.85 and 0.90. The insets show the enlarged region for the left side.

FORCE REQUIRED TO MOVE THE FLEXIBLE STRIP UP SURFACE OF HORIZONTAL CYLINDER

A differential equation of the motion of a flexible incompressible strip with a rectangular cross-section curve along the inner rough surface of a horizontal cylinder has been derived in the article. The strip moves up perpendicularly to the generatrix of the cylinder with a given constant speed, that is, the trajectory of movement is the cross-section curve of the cylinder. The equation takes into account the weight forces of the strip and the friction force from the resulting pressure force of the strip on the surface, as well as the additional friction force depending on the compression of the strip and the angle of its coverage with the cylinder. An example for a circular cylinder is considered. The differential equation is solved, the necessary force for pushing the strip is found. The graphs illustrating the influence of different factors on the pushing force of a strip at a given angle of its coverage are constructed.

Hall resistance anomalies in the integer and fractional quantum Hall regime

Experimental evidence of resistance anomalies in the high-mobility two-dimensional electron gas (2DEG) formed in the GaAs/AlGaAs heterostructure, in the integer and fractional quantized Hall regime, is shown. The data complement to a good approximation the semianalytic calculations used to describe the formation of integral and fractional incompressible strips. The widths of current-carrying channels were calculated by incorporating the screening properties of the 2DEG and the effect of a magnetic field in the perpendicular mode. The many-body effects of the composite fermions are taken into consideration for the energy gap for the fractional states. It is shown that incompressible strips at the edges for both integer and fractional filling factors coexist in their evanescent phase for a particular range of magnetic fields, resulting in overshoot effects at the Hall resistance. Specifically, anomalous Hall resistances were noticed for filling factors $\ensuremath{\nu}$ = $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{8}{3}$, 3, $\frac{10}{3}$, $\frac{7}{2}$, and 5. This effect is explained and discussed using the screening theory.

Electrostatic theory of irregular conductance oscillations through a quantum constriction under quantized Hall conditions

Abstract We propose a semi-classical explanation to the conductance oscillations ( δ G ) observed at gate-controlled quantum dots under quantized Hall conditions. Our approach is based on the self-consistent screening theory of the quantum Hall effect and δ G originates either from the interference of electrons or from charging effects. In the interference regime electrons acquire an Aharonov–Bohm phase while propagating coherently along the evanescent incompressible stripes at a particular filling factor, where the stripes enclose a magnetic flux. The charging regime finds its ground in Coulomb blocked, where the transport electron generates an additional energy scale to conductance oscillations. Combining these two regimes, we found that the period of conductance oscillations in magnetic field ( Δ B ) may increase, decrease or remain constant at different filling factors, as observed in the experiments. In addition it is shown that, a gate potential can shift the incompressible stripe along the gated edge spatially, thereby changing the enclosed flux, and can induce δ G with a period Δ V g . We demonstrate that our formulation is able to generate the superimposed Δ B - Δ V g interferences in stripe or checkerboard patterns in agreement with experiments. We can account for further salient and peculiar features of experimental observations for example the surprising absence of oscillations at certain filling factors.

Self-Consistent Theory of Screening and Transport in Narrow, Translation-Invariant Hall Bars under the Conditions of the Integer Quantum-Hall-Effect

We summarize and discuss a self-consistent screening and magneto-transport theory, developed to understand the results of scanning-force-microscope experiments on the current distribution in a two-dimensional electron system (2DES), located in a narrow Hall bar under the conditions of the integer quantum Hall effect (IQHE) and its breakdown. The theory explains why, at low temperatures, at certain intervals of the applied perpendicular magnetic field, the current density is confined to “incompressible stripes” (ISs), in which a fixed number of Landau levels is occupied, and the longitudinal and Hall resistivity assume quantized values. The theory also explains, how the position and shape of these ISs depend on magnetic field and temperature, and why the confinement of the current density on these ISs leads to precisely quantized values of the macroscopic longitudinal and Hall resistance. The theory, which leads, at high temperatures, to the well known Drude results for current density and resistance, shows that the IQHE is a consequence of the peculiar screening properties of a 2DES in a strong magnetic field at low temperatures, and that it can be understood without assumptions about special localization effects.

Probing the breakdown of topological protection: Filling-factor-dependent evolution of robust quantum Hall incompressible phases

The integer quantum Hall (QH) effects characterized by topologically quantized and nondissipative transport are caused by an electrically insulating incompressible phase that prevents backscattering between chiral metallic channels. We probed the incompressible area susceptible to the breakdown of topological protection using a scanning gate technique incorporating nonequilibrium transport. The obtained pattern revealed the filling-factor ($\nu$)-dependent evolution of the microscopic incompressible structures located along the edge and in the bulk region. We found that these specific structures, respectively attributed to the incompressible edge strip and bulk localization, show good agreement in terms of $\nu$-dependent evolution with a calculation of the equilibrium QH incompressible phases, indicating the robustness of the QH incompressible phases under the nonequilibrium condition. Further, we found that the $\nu$ dependency of the incompressible patterns is, in turn, destroyed by a large imposed current during the deep QH effect breakdown. These results demonstrate the ability of our method to image the microscopic transport properties of a topological two-dimensional system.

Tuning single-electron charging and interactions between compressible Landau level islands in graphene

Interacting and tunable quantum dots (QDs) have been extensively exploited in condensed matter physics and quantum information science. Using a low-temperature scanning tunneling microscope (STM), we both create and directly image a new type of coupled QD system in graphene, a highly interacting quantum relativistic system with tunable density. Using detailed scanning tunneling spectroscopy (STS) measurements, we show that Landau quantization inside a potential well enables novel electron confinement via the incompressible strips between partially filled Landau levels (LLs), forming isolated and concentric LL QDs. By changing the charge density and the magnetic field we can tune continuously between single- and double-concentric LL QD systems within the same potential well. In the concentric QD regime, single-electron charging peaks of the two dots intersect, displaying a characteristic avoidance pattern. At moderate fields, we observe an unconventional avoidance pattern that differs significantly from that observed in capacitively coupled double-QD systems. We find that we can reproduce in detail this anomalous avoidance pattern within the framework of the electrostatic double-QD model by replacing the capacitive interdot coupling with a phenomenological charge-counting system in which charges in the inner concentric dot are counted in the total charge of both islands. The emergence of such strange forms of interdot coupling in a single potential well, together with the ease of producing such charge pockets in graphene and other two-dimensional (2D) materials, reveals an intriguing testbed for the confinement of 2D electrons in customizable potentials.

The self-consistent quantum-electrostatic problem in strongly non-linear regime

The self-consistent quantum-electrostatic (also known as Poisson-Schrödinger) problem is notoriously difficult in situations where the density of states varies rapidly with energy. At low temperatures, these fluctuations make the problem highly non-linear which renders iterative schemes deeply unstable. We present a stable algorithm that provides a solution to this problem with controlled accuracy. The technique is intrinsically convergent even in highly non-linear regimes. We illustrate our approach with both a calculation of the compressible and incompressible stripes in the integer quantum Hall regime as well as a calculation of the differential conductance of a quantum point contact geometry. Our technique provides a viable route for the predictive modeling of the transport properties of quantum nanoelectronics devices.

Current distribution in narrow translation-invariant quantum-Hall-systems with lateral density modulation

A previously developed self-consistent screening and magneto-transport theory for laterally confined, translation-invariant quantum-Hall-systems is applied to two-dimensional electron systems created by a donor sheet with a lateral density modulation. The previous calculations, assuming a homogeneous donor charge density, could explain experimental results on the spatial distribution of an applied source-drain-current, and the resulting Hall potential, only for the ‘edge-dominated’ low-magnetic-field part of a quantum-Hall-plateau, where the current flows through incompressible stripes near the edges. For the high-magnetic-field regime of the plateau, they predicted current flow only in a narrow stripe in the center of the sample, whereas the experiments found current in a wide region of its bulk. Assuming a suitably modulated donor charge density, we can avoid this discrepancy, and we obtain a strong dependence of the distribution of the applied current on magnetic field, lattice temperature, and the current-strength.

Tunneling Spectroscopy of Quantum Hall States in Bilayer Graphene p-n Junctions.

We report tunneling transport in spatially controlled networks of quantum Hall (QH) edge states in bilayer graphene. By manipulating the separation, location, and spatial span of QH edge states via gate-defined electrostatics, we observe resonant tunneling between copropagating QH states across incompressible strips. Employing spectroscopic tunneling measurements and an analytical model, we characterize the energy gap, width, density of states, and compressibility of the QH edge states with high precision and sensitivity within the same device. The capability to engineer the QH edge network also provides an opportunity to build future quantum electronic devices with electrostatic manipulation of QH edge states, supported by rich underlying physics.

Counterflowing edge current and its equilibration in quantum Hall devices with sharp edge potential: Roles of incompressible strips and contact configuration

We report the observation of counterflowing edge current in InAs quantum wells which leads to the breakdown of quantum Hall (QH) effects at high magnetic fields. Counterflowing edge channels arise from the Fermi-level pinning of InAs and the resultant sharp edge potential with downward bending. By measuring the counterflow conductance for varying edge lengths, we determine the effective number $\langle N_\text{C} \rangle$ of counterflowing modes and their equilibration length $\lambda_\text{eq}$ at bulk integer filling factor $\nu = 1$--$4$. $\lambda_\text{eq}$ increased exponentially with magnetic field $B$, reaching $200~\mu$m for $\nu = 4$ at $B \geq 7.6$~T. Our data reveal important roles of the innermost incompressible strip with even filling in determining $\langle N_\text{C} \rangle$ and $\lambda_\text{eq}$ and the impact of the contact configuration on the QH effect breakdown. Our results show that counterflowing edge channels manifest as transport anomalies only at high fields and in short edges. This in turn suggests that, even in the integer QH regime, the actual microscopic structure of edge states can differ from that anticipated from macroscopic transport measurements, which is relevant to various systems including atomic-layer materials.

Classical and quantum capacitances calculated locally considering a two-dimensional Hall bar

We propose a semi-classical explanation to the conductance oscillations (δG) observed at gate-controlled quantum dots under quantized Hall conditions. Our approach is based on the self-consistent screening theory of the quantum Hall effect and δG originates either from the interference of electrons or from charging effects. In the interference regime electrons acquire an Aharonov–Bohm phase while propagating coherently along the evanescent incompressible stripes at a particular filling factor, where the stripes enclose a magnetic flux. The charging regime finds its ground in Coulomb blocked, where the transport electron generates an additional energy scale to conductance oscillations. Combining these two regimes, we found that the period of conductance oscillations in magnetic field (ΔB) may increase, decrease or remain constant at different filling factors, as observed in the experiments. In addition it is shown that, a gate potential can shift the incompressible stripe along the gated edge spatially, thereby changing the enclosed flux, and can induce δG with a period ΔVg. We demonstrate that our formulation is able to generate the superimposed ΔB - ΔVg interferences in stripe or checkerboard patterns in agreement with experiments. We can account for further salient and peculiar features of experimental observations for example the surprising absence of oscillations at certain filling factors.

Single-electron transport through quantum point contact

Here, we employ a numerical approach to investigate the transport and conductance characteristics of a quantum point contact. A quantum point contact is a narrow constriction of a width comparable to the electron wavelength defined in a two-dimensional electron gas (2DEG) by means of split-gate or etching technique. Their properties have been widely investigated in the experiments. In our study, we define a quantum Hall based split-gate quantum point contact with standard gate geometry. Firstly, we obtain the spatial distribution of incompressible strips (current channels) by applying a self consistent Thomas-Fermi method to a realistic heterostructure under quantized Hall conditions. Later, time-dependent Schrodinger equation is solved for electrons injected in the current channels. The transport characteristics and time-evolutions are analyzed in the integer filling factor regime (ν = 1) with the single electron density. The results confirm that the current direction in a realistic quantum point contact can be controllable with the external interventions.

Formation of spin droplet atν=52in an asymmetric quantum dot under quantum Hall conditions

In this work, a quantum dot that is defined asymmetrically by electrostatic means induced on a GaAs/AlGaAs heterostructure is investigated to unravel the effect of geometric constrains on the formation of spin droplets under quantised Hall conditions. The incompressibility of exciting nu = 5/2 state is explored by solving the Schrodinger equation within spin density functional theory, where the confinement potential is obtained self-consistently utilising the Thomas-Fermi approximation. Our numerical investigations show that the spatial distribution of the nu = 2 incompressible strips and electron occupation in the second lowest Landau level considerably differ from the results of the laterally symmetric quantum dots. Our findings yield two important consequences, first the incompressibility of the intriguing nu = 5/2 state is strongly affected by the asymmetry, and second, since the Aharonov-Bohm interference patterns depend on the velocity of the particles, asymmetry yields an additional parameter to adjust the oscillation period, which imposes a boundary condition dependency in observing quasi-particle phases.

The local nature of incompressibility of quantum Hall effect

Since the experimental realization of the integer quantum Hall effect in a two-dimensional electron system, the interrelation between the conductance quantization and the topological properties of the system has been investigated. Assuming that the two-dimensional electron system is described by a Bloch Hamiltonian, system is insulating in the bulk of sample throughout the quantum Hall plateau due to a magnetic field induced energy gap. Meanwhile, the system is conducting at the edges resembling a 2+1 dimensional topological insulator without time-reversal symmetry. Here, by our magneto-transport measurements performed on GaAs/AlGaAs high purity Hall bars with two inner contacts we show that incompressible strips formed at the edges result in Hall quantization, even if the bulk is compressible. Consequently, the relationship between the quantum Hall effect and topological bulk insulator breaks for specific field intervals within the plateaus. The measurement of conducting bulk, strongly challenges all existing single-particle theories.

Competition between Kondo Screening and Quantum Hall Edge Reconstruction.

We report on a Kondo correlated quantum dot connected to two-dimensional leads where we demonstrate the renormalization of the g factor in the pure Zeeman case. i.e., for magnetic fields parallel to the plane of the quantum dot. For the same system, we study the influence of orbital effects by investigating the quantum Hall regime; i.e., a perpendicular magnetic field is applied. In this case an unusual behavior of the suppression of the Kondo effect and of the split zero-bias anomaly is observed. The splitting decreases with magnetic field and shows discontinuous changes that are attributed to the intricate interplay between Kondo screening and the quantum Hall edge structure originating from electrostatic screening. This edge structure, made up of compressible and incompressible stripes, strongly affects the Kondo temperature of the quantum dot and thereby influences the renormalized g factor.

Circularly polarized near-field scanning optical microscope for investigations of edge states of a two-dimensional electron system

We report on investigations of the quantum Hall chiral edge states using a near-field scanning optical microscope that enables us to irradiate circularly polarized light from the probe tip with spatial resolution below the diffraction limit. We have found a clear evidence for the formation of spin-split incompressible strips near the edge of a two-dimensional electron system in high magnetic fields.