We demonstrate an innovative quantum Hall circuit with variable geometry employing the moveable electrostatic potential induced by a biased atomic force microscope tip. We exploit this additional degree of freedom to identify the microscopic mechanisms that allow two co-propagating edge channels to equilibrate their charge imbalance. Experimental results are compared with tight-binding simulations based on a realistic model for the disorder potential. This work provides also an experimental realization of a beam mixer between co-propagating edge channels, a still elusive building block of a recently proposed new class of quantum interferometers.
Charge equilibration between quantum-Hall edge states can be studied to reveal geometric structure of edge channels not only in the integer quantum Hall (IQH) regime but also in the fractional quantum Hall (FQH) regime particularly for hole-conjugate states. Here we report on a systematic study of charge equilibration in both IQH and FQH regimes by using a generalized Hall bar, in which a quantum Hall state is nested in another quantum Hall state with different Landau filling factors. This provides a feasible way to evaluate equilibration in various conditions even in the presence of scatterin…
By using four-terminal configurations, we investigated the dependence of longitudinal and diagonal resistances of a graphene p-n interface on the quantum-Hall edge-state equilibration position. The resistance of a p-n device in our four-terminal scheme is asymmetric with respect to the zero point where the filling factor ($ν$) of the entire graphene vanishes. This resistance asymmetry is caused by the chiral-direction-dependent change of the equilibration position and leads to a deeper insight into the equilibration process of the quantum-Hall edge states in a bipolar graphene system.
Conductance of the edge modes as well as conductance across the co-propagating edge modes around the ν= 4/3, 5/3 and 2 quantum Hall states are measured by individually exciting the modes. Temperature dependent equilibration rates of the outer unity conductance edge mode are presented for different filling fractions. We find that the equilibration rate of the outer unity conductance mode at ν= 2 is higher and more temperature sensitive compared to the mode at fractional filling 5/3 and 4/3. At lowest temperature, equilibration length of the outer unity conductance mode tends to saturate with lo…
Equilibration of quantum Hall edges is studied in a high quality dual gated bilayer graphene device in both unipolar and bipolar regime when all the degeneracies of the zero energy Landau level are completely lifted. We find that in the unipolar regime when the filling factor under the top gate region is higher than the back gate filling factor, the equilibration is partial based on their spin polarization. However, the complete mixing of the edge states is observed in the bipolar regime irrespective of their spin configurations due to the Landau level collapsing at the sharp pn junction in ou…
We experimentally study equilibration between edge states, co-propagating at the edge of the fractional quantum Hall liquid, at high initial imbalances. We find an anomalous increase of the conductance between the fractional edge states at the filling factor $ν=2/5$ in comparison with the expected one for the model of independent edge states. We conclude that the model of independent fractional edge states is not suitable to describe the experimental situation at $ν=2/5$.
Fractional conductance is measured by partitioning $ν= 1$ edge state using gate-tunable fractional quantum Hall (FQH) liquids of filling 1/3 or 2/3 for current injection and detection. We observe two sets of FQH plateaus 1/9, 2/9, 4/9 and 1/6, 1/3, 2/3 at low and high magnetic field ends of the $ν= 1$ plateau respectively. The findings are explained by magnetic field dependent equilibration of three FQH edge modes with conductance $e^2/3h$ arising from edge reconstruction. The results reveal remarkable enhancement of the equilibration lengths of the FQH edge modes with increasing field.
The equilibration between quantum Hall edge modes is known to depend on the disorder potential and the steepness of the edge. Modern samples with higher mobilities and setups with lower electron temperatures call for a further exploration of the topic. We develop a framework to systematically measure and analyze the equilibration of many (up to 8) integer edge modes. Our results show that spin-selective coupling dominates even for non-neighboring channels with parallel spin. Changes in magnetic field and bulk density let us control the equilibration until it is almost completely suppressed and…
We probe quantum Hall effect in a tunable 1-D lateral superlattice (SL) in graphene created using electrostatic gates. Lack of equilibration is observed along edge states formed by electrostatic gates inside the superlattice. We create strong local electric field at the interface of regions of different charge densities. Crossed electric and magnetic fields modify the wavefunction of the Landau Levels (LLs) - a phenomenon unique to graphene. In the region of copropagating electrons and holes at the interface, the electric field is high enough to modify the Landau levels resulting in increased …
Hole-conjugate states of the fractional quantum Hall effect host counter-propagating edge channels which are thought to exchange charge and energy. These exchanges have been the subject of extensive theoretical and experimental works; in particular, it is yet unclear if the presence of integer quantum Hall edge channels stemming from fully filled Landau levels affects heat equilibration along the edge. In this letter, we present heat transport measurements in quantum Hall states of graphene demonstrating that the integer channels can strongly equilibrate with the fractional ones, leading to ma…
Two-dimensional topological insulators, and in particular quantum Hall states, are characterized by an insulating bulk and a conducting edge. Fractional states may host both downstream (dictated by the magnetic field) and upstream propagating edge modes, which leads to complex transport behavior. Here, we combine two measurement techniques, local noise thermometry and thermal conductance, to study thermal properties of states with counter-propagating edge modes. We find that, while charge equilibration between counter-propagating edge modes is very fast, the equilibration of heat is extremely …
Since the charged mode is much faster than the neutral modes on quantum Hall edges at large filling factors, the edge may remain out of equilibrium in thermal conductance experiments. This sheds light on the observed imperfect quantization of the thermal Hall conductance at $ν=8/3$ and can increase the observed thermal conductance by two quanta at $ν=8/5$. Under certain unlikely but not impossible assumptions, this might also reconcile the observed thermal conductance at $ν=5/2$ with not only the PH-Pfaffian order but also the anti-Pfaffian order.
The equilibration length between spin-polarized edge states in the Quantum Hall regime is measured as a function of a gate voltage applied to an electrode on top of the edge channels. Reproducible fluctuations in the coupling are observed and interpreted as a mesoscopic fingerprint of single spin-flip scatterers which are turned on and off. A model to analyze macroscopic edge state coupling in terms of individual scatterers is developed, and characteristic values for these scatterers in our samples are extracted. For all samples investigated, the distance between spin-flip scatterers lies betw…
We realize p-p'-p junctions in few-layer black phosphorus (BP) devices, and use magneto-transport measurements to study the equilibration and transmission of edge states at the interfaces of regions with different charge densities. We observe both full equilibration, where all edge channels equilibrate and are equally partitioned at the interfaces, and partial equilibration, where only equilibration only takes place among modes of the same spin polarization. Furthermore, the inner p'-region with low-doping level in the junction can function as a filter for highly doped p-regions which demonstr…
Transport through edge-channels is responsible for conduction in quantum Hall (QH) phases. Topology dictates quantization of both charge and thermal transport coefficients. These turn out to approach robust quantized values when incoherent equilibration processes become dominant. Here, we report on measurements of both electrical and thermal conductances of integer and fractional quantum Hall (FQH) phases, realized in hBN encapsulated graphite gated bilayer graphene devices. Remarkably, for the complex edge at filling factors $ν=5/3$ and $ν=8/3$, which correspond to the paradigmatic hole-conju…
Ohmic contacts are crucial elements of electron optics that have not received a clear theoretical description yet. We propose a model of an Ohmic contact as a piece of metal of the finite capacitance $C$ attached to a quantum Hall edge. It is shown that charged quantum Hall edge states may have weak coupling to neutral excitations in an Ohmic contact. Consequently, despite being a reservoir of neutral excitations, an Ohmic contact is not able to efficiently equilibrate edge states if its temperature is smaller than $\hbarΩ_c$, where $Ω_c$ is the inverse RC time of the contact. This energy scal…
We report an observation of conductance fuctuations (CFs) in the bipolar regime of quantum hall (QH) plateaus in graphene (p-n-p/n-p-n) devices. The CFs in the bipolar regime are shown to decrease with increasing bias and temperature. At high temperature (above 7 K) the CFs vanishes completely and the flat quantized plateaus are recovered in the bipolar regime. The values of QH plateaus are in theoretical agreement based on full equilibration of chiral channels at the p-n junction. The amplitude of CFs for different filling factors follows a trend predicted by the random matrix theory. Althoug…
We report on transport measurements of dual-gated, single-layer graphene devices in the quantum Hall regime, allowing for independent control of the filling factors in adjoining regions. Progress in device quality allows us to study scattering between edge states when the four-fold degeneracy of the Landau level is lifted by electron correlations, causing edge states to be spin and/or valley polarized. In this new regime, we observe a dramatic departure from the equilibration seen in more disordered devices: edge states with opposite spins propagate without mixing. As a result, the degree of e…
When two separately contacted quantum Hall (QH) edge channels are brought into interaction, they can equilibrate their imbalance via scattering processes. In the present work we use a tunable QH circuit to implement a junction between co-propagating edge channels whose length can be controlled with continuity. Such a variable device allows us to investigate how current-voltage characteristics evolve when the junction length d is changed. Recent experiments with fixed geometry reported a significant reduction of the threshold voltage for the onset of photon emission, whose origin is still under…
We have demonstrated the millimeter-scale fabrication of monolayer epitaxial graphene $p-n$ junction devices using simple ultraviolet photolithography, thereby significantly reducing device processing time compared to that of electron beam lithography typically used for obtaining sharp junctions. This work presents measurements yielding nonconventional, fractional multiples of the typical quantized Hall resistance at $ν=2$ ($R_H\approx 12906 Ω$) that take the form: $\frac{a}{b}R_H$. Here, $a$ and $b$ have been observed to take on values such 1, 2, 3, and 5 to form various coefficients of $R_H$…
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 $λ_\text{eq}$ at bulk integer filling factor $ν= 1$--$4$. $λ_\text{eq}$ increased exponentially with magnetic field $B$…
Ballistic-to-diffusive transition in engineered counter-propagating quantum Hall channels
Université Paris-Saclay, CEA, CNRS, SPEC（巴黎萨克雷大学、法国原子能和替代能源委员会、法国国家科学研究中心、物理谱与现象实验室）；Research Center for Electronic and Optical Materials, National Institute for Materials Science（物质材料研究所电子与光学材料研究中心）；Research Center for Materials Nanoarchitectonics, National Institute for Materials Science（物质材料研究所材料纳米架构研究中心）；Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité（巴黎高等师范学院物理实验室、巴黎高等师范学院、巴黎文理研究大学、法国国家科学研究中心、索邦大学、巴黎西岱大学）
Using an innovative combination of a quasi-Corbino sample geometry and the cross-gate technique, we have developed a method that enables us to separately contact single edge channels in the quantum Hall regime and investigate equilibration among them. Performing 4-point resistance measurements, we directly obtain information on the energetic and geometric structure of the edge region and the equilibration-length for current transport across the Landau- as well as the spin-gap. Based on an almost free choice in the number of participating edge channels and their interaction-length a systematic …
Thermal conductance has emerged as a powerful probe of topological order in the quantum Hall effect and beyond. The interpretation of experiments crucially depends on the ratio of the sample size and the equilibration length, on which energy exchange among contra-propagating chiral modes becomes significant. We show that at low temperatures the equilibration length diverges as $1/T^2$ for almost all Abelian and non-Abelian topological orders. A faster $1/T^4$ divergence is present on the edges of the non-Abelian PH-Pfaffian and negative-flux Read-Rezayi liquids. We address experimental consequ…
We study equilibration of quantum Hall edge states at integer filling factors, motivated by experiments involving point contacts at finite bias. Idealising the experimental situation and extending the notion of a quantum quench, we consider time evolution from an initial non-equilibrium state in a translationally invariant system. We show that electron interactions bring the system into a steady state at long times. Strikingly, this state is not a thermal one: its properties depend on the full functional form of the initial electron distribution, and not simply on the initial energy density. F…
We study electron transport through a multichannel fractional quantum Hall edge in the presence of both interchannel interaction and random tunneling between channels, with emphasis on the role of contacts. The prime example in our discussion is the edge at filling factor 2/3 with two counterpropagating channels. Having established a general framework to describe contacts to a multichannel edge as thermal reservoirs, we particularly focus on the line-junction model for the contacts and investigate incoherent charge transport for an arbitrary strength of interchannel interaction beneath the con…
We experimentally study equilibration across the sample edge at high fractional filling factors 4/3, 5/3 under experimental conditions, which allow us to obtain high imbalance conditions. We find a lack of the full equilibration across the edge even in the flat-band situation, where no potential barrier survives at the sample edge. We interpret this result as the manifestation of complicated edge excitation structure at high fractional filling factors 4/3, 5/3. Also, a mobility gap in the $ν_c=1$ incompressible strip is determined in normal and tilted magnetic fields.
We consider electrical and thermal equilibration of the edge modes of the Anti-Pfaffian quantum Hall state at $ν=5/2$ due to tunneling of the Majorana edge mode to trapped Majorana zero modes in the bulk. Such tunneling breaks translational invariance and allows scattering between Majorana and other edge modes in such a way that there is a parametric difference between the length scales for equilibration of charge and heat transport between integer and Bose mode on the one hand, and for thermal equilibration of the Majorana edge mode on the other hand. We discuss a parameter regime in which th…
We present direct experimental evidence for nonlocal transport in HgTe quantum wells in the quantum spin Hall regime, in the absence of any external magnetic field. The data conclusively show that the non-dissipative quantum transport occurs through edge channels, while the contacts lead to equilibration between the counter-propagating spin states at the edge. We show that the experimental data agree quantitatively with the theory of the quantum spin Hall effect.
Experimentally Motivated Order of Length Scales Affect Shot Noise
Weizmann Institute of Science（魏茨曼科学研究所）；Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University（特拉维夫大学雷蒙德和贝弗利萨克勒物理与天文学学院）；Indian Institute of Science Education and Research (IISER) Tirupati（提鲁帕蒂印度科学与教育研究院）
It is well established that 'density reconstruction' at the edge of a two dimensional electron gas takes place for 'hole-conjugate' states in the fractional quantum Hall effect (such as $ν=2/3, 3/5,$ etc.). Such reconstruction leads, after equilibration between counter propagating edge channels, to a downstream chiral current edge mode accompanied by upstream chiral neutral modes (carrying energy without net charge). Short equilibration length prevented thus far observation of the counter propagating current channels - the hallmark of density reconstruction. Here, we provide evidence for such …
We demonstrate a programmable quantum Hall circuit that implements a novel iterative voltage bisection scheme and allows obtaining any binary fraction $(k/2^n)$ of the fundamental resistance quantum $R_K/2=h/2e^2$. The circuit requires a number $n$ of bisection stages that only scales logarithmically with the precision of the fraction. The value of $k$ can be set to any integer between 1 and $2^n$ by proper gate configuration. The architecture exploits gate-controlled routing, mixing and equilibration of edge modes of robust quantum Hall states. The device does not contain {\em any} internal o…
Using a new MBE growth technique, we fabricate a two-dimensional electron system which is bent around an atomically sharp 90 degree corner. In the quantum Hall regime under tilted magnetic fields, we can measure equilibration between both co- and counter-propagating edge channels of arbitrary filling factor ratio. We present here 4-point magnetotransport characterization of the corner junction with filling factor combinations which can all be explained using the standard Landauer-Buttiker edge channel picture. The success of this description confirms the realization of a new type of quantum Ha…
A theory of transport in the quantum Hall regime is developed for separately contacted double-layer electron systems. Inter-layer tunneling provides a channel for equilibration of the distribution functions in the two layers at the edge states. Resistances and transresistances for various configurations of the electrodes are calculated as functions of the inter-layer tunneling amplitude. Induced current in one of the layer by a current in the other is calculated also. It is shown that reflection at the leads causes change in the results for some electrode configurations. The results obtained i…
Questions on the nature of edge reconstruction and "where does the current flow" in the quantum Hall effect (QHE) have been debated for years. Moreover, the recent observation of proliferation of "upstream" neutral modes in the fractional QHE raised doubts about the present models of edge channels. In this article we focus on hole-conjugate states, nu=2/3 and nu=3/5, and present a new picture of their edge reconstruction. For example, while the present model for nu=2/3 consists of a single downstream charge channel with conductance 2/3 e^2/h and an upstream neutral mode, we show that the curre…
We consider the absorption of microwaves by a quantum Hall droplet. We show that the number and velocities of charged edge modes can be directly measured from a droplet of known shape. In contrast to standard transport measurements, different edge equilibration regimes can be accessed in the same device. If there is a quantum point contact in the droplet, then quasiparticle properties, including braiding statistics, can be observed. Their effects are manifested as modulations of the microwave absorption spectrum that are, notably, first-order in the tunneling amplitude at the point contact.
Topological phases typically encode topology at the level of the single particle band structure. But a remarkable class of models shows that quantum anomalous Hall effects can be driven exclusively by interactions, while the parent non-interacting band structure is topologically trivial. Unfortunately, these models have so far relied on interactions that do not spatially decay and are therefore unphysical. We study a model of spinless fermions on a decorated honeycomb lattice. Using complementary methods, mean-field theory and exact diagonalization, we find a robust quantum anomalous Hall phas…
Reentrant integer quantum Hall (RIQH) states are believed to be correlated electron solid phases, though their microscopic description remains unclear. As bias current increases, longitudinal and Hall resistivities measured for these states exhibit multiple sharp breakdown transitions, a signature unique to RIQH states. We present spatially-resolved measurements of RIQH breakdown that indicate these breakdown signatures can be ascribed to a phase boundary between broken-down and unbroken regions, spreading chirally from source and drain contacts as a function of bias current and passing voltag…
Robust quantum anomalous Hall effect with spatially uncorrelated disorder
Imec；Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich（于利希研究中心彼得·格林伯格研究所（PGI-9））；JARA-Fundamentals of Future Information Technology, Jülich-Aachen Research Alliance, Forschungszentrum Jülich and RWTH Aachen University（亚琛-于利希研究联盟JARA未来信息技术基础，于利希研究中心和亚琛工业大学）；Department of Physics, Georgetown University（乔治城大学物理系）
We study the $ν=\frac{2}{k+2}$ quantum Hall states which are particle-hole conjugates of the $ν=\frac{k}{k+2}$ Read-Rezayi states. We find that equilibration between the different modes at the edge of such a state leads to an emergent SU(2)$_k$ algebra in the counter-propagating neutral sector. Heat flow along the edges of these states will be in the opposite direction of charge flow. In the $k=3$ case, which may be relevant to $ν=2+{2/5}$, the thermal Hall conductance and the exponents associated with quasiparticle and electron tunneling distinguish this state from competing states such as th…
We investigate how the criticality of the quantum Hall plateau transition in disordered graphene differs from those in the ordinary quantum Hall systems, based on the honeycomb lattice with ripples modeled as random hoppings. The criticality of the graphene-specific n=0 Landau level is found to change dramatically to an anomalous, almost exact fixed point as soon as we make the random hopping spatially correlated over a few bond lengths. We attribute this to the preserved chiral symmetry and suppressed scattering between K and K' points in the Brillouin zone. The results suggest that a fixed p…
We have studied quantized transport in HgTe wells with inverted band structure corresponding to the two-dimensional topological insulator phase (2D TI) with locally-controlled density allowing n-p-n and n-2D TI-n junctions. The resistance reveals the fractional plateau $2h/e^{2}$ in n-p-n regime in the presence of the strong perpendicular magnetic field. We found that in n-2D TI-n regime the plateaux in resistance in not universal and results from the edge state equilibration at the interface between chiral and helical edge modes. We provided the simple model describing the resistance quantiza…
Quantum transport properties in quantum Hall wires in the presence of spatially correlated random potential are investigated numerically. It is found that the potential correlation reduces the localization length associated with the edge state, in contrast to the naive expectation that the potential correlation increases it. The effect appears as the sizable shift of quantized conductance plateaus in long wires, where the plateau transitions occur at energies much higher than the Landau band centers. The scale of the shift is of the order of the strength of the random potential and is insensit…
Quantum transport properties in quantum Hall wires in the presence of spatially correlated disordered magnetic fields are investigated numerically. It is found that the correlation drastically changes the transport properties associated with the edge state, in contrast to the naive expectation that the correlation simply reduces the effect of disorder. In the presence of correlation, the separation between the successive conductance plateau transitions becomes larger than the bulk Landau level separation determined by the mean value of the disordered magnetic fields. The transition energies co…
We propose an experimental scheme to observe spin Hall effects with cold atoms in a light induced gauge potential. Under an appropriate configuration, the cold atoms moving in a spatially varying laser field experience an effective spin-dependent gauge potential. Through numerical simulation, we demonstrate that such a gauge field leads to observable spin Hall currents under realistic conditions. We also discuss the quantum spin Hall state in an optical lattice.
An experiment to probe spin-polarized currents in the quantum Hall regime is suggested that takes advantage of the large Zeeman-splitting in the paramagnetic diluted magnetic semiconductor zinc manganese selenide (Zn$_{1-x}$Mn$_x$Se). In the proposed experiment spin-polarized electrons are injected by ZnMnSe-contacts into a gallium arsenide (GaAs) two-dimensional electron gas (2DEG) arranged in a Hall bar geometry. We calculated the resulting Hall resistance for this experimental setup within the framework of the Landauer-Büttiker formalism. These calculations predict for 100% spininjection th…
The discovery of the Quantum Spin Hall state, and topological insulators in general, has sparked strong experimental efforts. Transport studies of the Quantum Spin Hall state confirmed the presence of edge states, showed ballistic edge transport in micron-sized samples and demonstrated the spin polarization of the helical edge states. While these experiments have confirmed the broad theoretical model, the properties of the QSH edge states have not yet been investigated on a local scale. Using Scanning Gate Microscopy to perturb the QSH edge states on a sub-micron scale, we identify well-locali…
The fate of the current carrying states of a quantum Hall system is considered in the situation when the disorder strength is increased and the transition from the quantum Hall liquid to the Hall insulator takes place. We investigate a two-dimensional lattice model with spatially correlated disorder potentials and calculate the density of states and the localization length either by using a recursive Green function method or by direct diagonalization in connection with the procedure of level statistics. From the knowledge of the energy and disorder dependence of the localization length and the…
Both topological crystalline insulators surfaces and graphene host multi-valley massless Dirac fermions which are not pinned to a high-symmetry point of the Brillouin zone. Strain couples to the low-energy electrons as a time-reversal invariant gauge field, leading to the formation of pseudo-Landau levels (PLL). Here we study periodic pseudo-magnetic fields originating from strain superlattices. We study the low-energy Dirac PLL spectrum induced by the strain superlattice and analyze the effect of various polarized states. Through self-consistent Hartree-Fock calculations we establish that, du…
We have developed a device fabrication process to pattern graphene into nanostructures of arbitrary shape and control their electronic properties using local electrostatic gates. Electronic transport measurements have been used to characterize locally gated bipolar graphene $p$-$n$-$p$ junctions. We observe a series of fractional quantum Hall conductance plateaus at high magnetic fields as the local charge density is varied in the $p$ and $n$ regions. These fractional plateaus, originating from chiral edge states equilibration at the $p$-$n$ interfaces, exhibit sensitivity to inter-edge backsc…
We present transport measurements on a lateral p-n junction in an inverted InAs/GaSb double quantum well at zero and nonzero perpendicular magnetic fields. At a zero magnetic field, the junction exhibits diodelike behavior in accordance with the presence of a hybridization gap. With an increasing magnetic field, we explore the quantum Hall regime where spin-polarized edge states with the same chirality are either reflected or transmitted at the junction, whereas those of opposite chirality undergo a mixing process, leading to full equilibration along the width of the junction independent of sp…
