The microscopic nature of localization in the quantum Hall effect

Enhancing impacts of mesoscale eddies on Southern Ocean temperature variability and extremes

As a major sink of anthropogenic heat and carbon, the Southern Ocean experienced pronounced warming with increasing extreme temperature events over the past decades. Mesoscale eddies that strongly influence the uptake, redistribution, and storage of heat in the ocean are expected to play important roles in these changes, yet observational evidence remains limited. Here, we employ a comprehensive analysis of over 500,000 historical hydrographic profile measurements combined with satellite-based eddy observations to show enhanced thermal eddy imprints in the Southern Ocean. Our observations reveal that anticyclonic (cyclonic) eddies are responsible for nearly half of the subsurface high (low)-temperature extremes detected, although only 10% of the profiles are located in eddy interiors. Over the past decade (2006 to 2019), both mean and extreme temperature anomalies within eddies in the Antarctic Circumpolar Current increased significantly, promoting the rise in subsurface ocean temperature variability. This enhanced role of eddies is likely a result of enhanced eddy pumping due to the increase in eddy intensity and ocean stratification caused by ocean warming. Our analysis underscores the crucial role of eddies in amplifying ocean temperature variability and extremes, with their effects expected to be even more pronounced as global warming persists.

Multifractal Conductance Fluctuations in High-Mobility Graphene in the Integer Quantum Hall Regime

We present the first experimental evidence for the multifractality of a transport property at a topological phase transition. In particular, we show that conductance fluctuations display multifractality at the integer quantum Hall plateau-to-plateau transitions in high-mobility mesoscopic graphene devices. The multifractality gets rapidly suppressed as the chemical potential moves away from these critical points. Our combination of experimental study and multifractal analysis provides a novel method for probing the criticality of wave functions at phase transitions in mesoscopic systems, and quantum criticality in several condensed-matter systems.

Microwave Dynamical Conductivity in the Quantum Hall Regime

Dynamical conductivity contains information of dissipative and nondissipative processes induced by ac-electric fields. In the integer quantum Hall (QH) effect where the nondissipative Hall current is the most prominent feature, its robustness is assured by localized states within the Landau levels. We establish a noncontact method with a circular cavity resonator and detect the real and imaginary parts of the longitudinal and Hall conductivities at a microwave frequency in magnetic fields. The conventional Shubnikov-de Haas oscillations and QH plateaus are observed in the real parts of longitudinal and Hall conductivities, respectively, while periodic structures can be seen in the imaginary parts which are scaled by the QH filling factor. The latter originates from intra-Landau level transitions between different orbital angular momenta. The results demonstrate that the dynamical conductivity measurement provides microscopic information which is not accessible by conventional static methods. The present noncontact method would pave the way to reveal the electron dynamics in other two-dimensional systems such as twisted bilayer graphene.

Domain Formation Driven by the Entropy of Topological Edge Modes

In this Letter we study interacting systems with spontaneous discrete symmetry breaking, where the degenerate symmetry-broken states are topologically distinct gapped phases. Edge modes appear at domain walls between the two topological phases. In the presence of a weak disorder field conjugate to the order parameter, we find that the entropy of the edge modes drives a thermal transition between a gapped uniform phase and a phase with a disorder-induced domain structure. We characterize this transition using a phenomenological Landau functional, and corroborate our conclusions with a concrete microscopic model. Finally, we discuss the possibilities of experimental signatures of this phase transition, and propose graphene-based moiré heterostructures as candidate materials in which such a phase transition can be detected.

Majorana Anyon Composites in Magneto-Photoluminescence Spectra of Natural Quantum Hall Puddles

In magneto-photoluminescence (magneto-PL) spectra of quasi two-dimensional islands (quantum dots) having seven electrons and Wigner−Seitz radius rs~1.5, we revealed a suppression of magnetic field (B) dispersion, paramagnetic shifts, and jumps of the energy of the emission components for filling factors ν > 1 (B < 10 T). Additionally, we observed B-hysteresis of the jumps and a dependence of all these anomalous features on rs. Using a theoretical description of the magneto-PL spectra and an analysis of the electronic structure of these dots based on the single-particle Fock−Darwin spectrum and many-particle configuration-interaction calculations, we show that these observations can be described by the rs-dependent formation of the anyon (magneto-electron) composites (ACs) involving single-particle states having non-zero angular momentum and that the anyon states observed involve Majorana modes (MMs), including zero-B modes having an equal number of vortexes and anti-vortexes, which can be considered as Majorana anyons. We show that the paramagnetic shift corresponds to a destruction of the equilibrium self-formed ν~5/2 AC by the external magnetic field and that the jumps and their hysteresis can be described in terms of Majorana qubit states controlled by B and rs. Our results show a critical role of quantum confinement in the formation of magneto-electrons and implies the liquid-crystal nature of fractional quantum Hall effect states, the Majorana anyon origin of the states having even ν, i.e., composite fermions, which provide new opportunities for topological quantum computing.

Correlated Double-Electron Additions at the Edge of a Two-Dimensional Electronic System

We create laterally large and low-disorder GaAs quantum-well-based quantum dots that act as small two-dimensional electron systems. We monitor tunneling of single electrons to the dots by means of capacitance measurements and identify single-electron capacitance peaks in the addition spectrum from occupancies of one up to thousands of electrons. The data show two remarkable phenomena in the Landau level filling factor range ν=2 to ν=5 in selective probing of the edge states of the dot: (i) Coulomb blockade peaks arise from the entrance of two electrons rather than one; (ii) at and near ν=5/2 and at fixed gate voltage, these double-height peaks appear uniformly in a magnetic field with a flux periodicity of h/2e, but they group into pairs at other filling factors.

Quantum-dot assisted spectroscopy of degeneracy-lifted Landau levels in graphene

Energy spectroscopy of strongly interacting phases requires probes which minimize screening while retaining spectral resolution and local sensitivity. Here, we demonstrate that such probes can be realized using atomic sized quantum dots bound to defects in hexagonal Boron Nitride tunnel barriers, placed at nanometric distance from graphene. With dot energies capacitively tuned by a planar graphite electrode, dot-assisted tunneling becomes highly sensitive to the graphene excitation spectrum. The spectra track the onset of degeneracy lifting with magnetic field at the ground state, and at unoccupied excited states, revealing symmetry-broken gaps which develop steeply with magnetic field - corresponding to Landé g factors as high as 160. Measured up to B = 33 T, spectra exhibit a primary energy split between spin-polarized excited states, and a secondary spin-dependent valley-split. Our results show that defect dots probe the spectra while minimizing local screening, and are thus exceptionally sensitive to interacting states.

Simultaneous voltage and current density imaging of flowing electrons in two dimensions

A variety of physical phenomena associated with nanoscale electron transport often results in non-trivial spatial voltage and current patterns, particularly in nonlocal transport regimes. While numerous techniques have been devised to image electron flows, the need remains for a nanoscale probe capable of simultaneously imaging current and voltage distributions with high sensitivity and minimal invasiveness, in a magnetic field, across a broad range of temperatures and beneath an insulating surface. Here we present a technique for spatially mapping electron flows based on a nanotube single-electron transistor, which achieves high sensitivity for both voltage and current imaging. In a series of experiments using high-mobility graphene devices, we demonstrate the ability of our technique to visualize local aspects of intrinsically nonlocal transport, as in ballistic flows, which are not easily resolvable via existing methods. This technique should aid in understanding the physics of two-dimensional electronic devices and enable new classes of experiments that image electron flow through buried nanostructures in the quantum and interaction-dominated regimes.

The interplay between Zeeman splitting and spin-orbit coupling in InAs nanowires

Coupling of the electron orbital motion and spin, i.e., spin-orbit coupling (SOC) leads to nontrivial changes in energy-level structures, giving rise to various spectroscopies and applications. The SOC in solids generates energy-band inversion or splitting under zero or weak magnetic fields, which is required for topological phases or Majorana fermions. Here, we examined the interplay between the Zeeman splitting and SOC by performing the transport spectroscopy of Landau levels (LLs) in indium arsenide nanowires under a strong magnetic field. We observed the anomalous Zeeman splitting of LLs, which depends on the quantum number of LLs as well as the electron spin. We considered that this observation was attributed to the interplay between the Zeeman splitting and the SOC. Our findings suggest an approach of generating spin-resolved chiral electron transport in nanowires.

Conductance quantization suppression in the quantum Hall regime

Conductance quantization is the quintessential feature of electronic transport in non-interacting mesoscopic systems. This phenomenon is observed in quasi one-dimensional conductors at zero magnetic field B, and the formation of edge states at finite magnetic fields results in wider conductance plateaus within the quantum Hall regime. Electrostatic interactions can change this picture qualitatively. At finite B, screening mechanisms in narrow, gated ballistic conductors are predicted to give rise to an increase in conductance and a suppression of quantization due to the appearance of additional conduction channels. Despite being a universal effect, this regime has proven experimentally elusive because of difficulties in realizing one-dimensional systems with sufficiently hard-walled, disorder-free confinement. Here, we experimentally demonstrate the suppression of conductance quantization within the quantum Hall regime for graphene nanoconstrictions with low edge roughness. Our findings may have profound impact on fundamental studies of quantum transport in finite-size, two-dimensional crystals with low disorder.

Conductance quantization is the hallmark of non-interacting confined systems. The authors show that the quantization in graphene nanoconstrictions with low edge disorder is suppressed in the quantum Hall regime. This is explained by the addition of new conductance channels due to electrostatic screening.

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.

Edge current quantization in the integer quantum Hall effect is understood to arise due to noninteracting electrons circulating an incompressible insulating bulk. Here, the authors evidence compressible metal-like bulk behaviour in GaAs/AlGaAs Hall bars consistent with electronic interactions.

Electronic and optical properties of graphene nanoribbons in external fields

A review work is done for the electronic and optical properties of graphene nanoribbons in magnetic, electric, composite, and modulated fields. Effects due to the lateral confinement, curvature, stacking, non-uniform subsystems and hybrid structures are taken into account. The special electronic properties, induced by complex competitions between external fields and geometric structures, include many one-dimensional parabolic subbands, standing waves, peculiar edge-localized states, width- and field-dependent energy gaps, magnetic-quantized quasi-Landau levels, curvature-induced oscillating Landau subbands, crossings and anti-crossings of quasi-Landau levels, coexistence and combination of energy spectra in layered structures, and various peak structures in the density of states. There exist diverse absorption spectra and different selection rules, covering edge-dependent selection rules, magneto-optical selection rule, splitting of the Landau absorption peaks, intragroup and intergroup Landau transitions, as well as coexistence of monolayer-like and bilayer-like Landau absorption spectra. Detailed comparisons are made between the theoretical calculations and experimental measurements. The predicted results, the parabolic subbands, edge-localized states, gap opening and modulation, and spatial distribution of Landau subbands, have been identified by various experimental measurements.

Wireless Majorana Bound States: From Magnetic Tunability to Braiding

We propose a versatile platform to investigate the existence of Majorana bound states (MBSs) and their non-Abelian statistics through braiding. This implementation combines a two-dimensional electron gas formed in a semiconductor quantum well grown on the surface of an s-wave superconductor with a nearby array of magnetic tunnel junctions (MTJs). The underlying magnetic textures produced by MTJs provide highly controllable topological phase transitions to confine and transport MBSs in two dimensions, overcoming the requirement for a network of wires. Obtained scaling relations confirm that various semiconductor quantum well materials are suitable for this proposal.

Signatures of single quantum dots in graphene nanoribbons within the quantum Hall regime

We report on the observation of periodic conductance oscillations near quantum Hall plateaus in suspended graphene nanoribbons. They are attributed to single quantum dots that are formed in the narrowest part of the ribbon, in the valleys and hills of a disorder potential. In a wide flake with two gates, a double-dot system's signature has been observed. Electrostatic confinement is enabled in single-layer graphene due to the gaps that are formed between the Landau levels, suggesting a way to create gate-defined quantum dots that can be accessed with quantum Hall edge states.

MgZnO/ZnO heterostructures with electron mobility exceeding 1 × 10(6) cm(2)/Vs

The inherently complex chemical and crystallographic nature of oxide materials has suppressed the purities achievable in laboratory environments, obscuring the rich physical degrees of freedom these systems host. In this manuscript we provide a systematic approach to defect identification and management in oxide molecular beam epitaxy grown MgZnO/ZnO heterostructures which host two-dimensional electron systems. We achieve samples displaying electron mobilities in excess of 1 × 10⁶ cm²/Vs. This data set for the MgZnO/ZnO system firmly establishes that the crystalline quality has become comparable to traditional semiconductor materials.

Fractional Quantum Hall States in Bilayer Graphene Probed by Transconductance Fluctuations

We have investigated fractional quantum Hall (QH) states in Bernal-stacked bilayer graphene using transconductance fluctuation measurements. A variety of odd-denominator fractional QH states with νQH → νQH + 2 symmetry, as previously reported, are observed. However, surprising is that also particle-hole symmetric states are clearly resolved in the same measurement set. We attribute their emergence to the reversal of orbital states in the octet level scheme induced by a strong local charge imbalance, which can be captured by the transconductance fluctuations. Also the even-denominator fractional QH state at filling -1/2 is observed. However, contrary to a previous study on a suspended graphene layer [ Ki et al. Nano Lett. 2014, 14 , 2135 ], the particle-hole symmetric state at filling 1/2 is detected as well. These observations suggest that the stability of both odd and even denominator fractional QH states is very sensitive to local transverse electric fields in bilayer graphene.

Shot-noise evidence of fractional quasiparticle creation in a local fractional quantum Hall state

We experimentally identify fractional quasiparticle creation in a tunneling process through a local fractional quantum Hall (FQH) state. The local FQH state is prepared in a low-density region near a quantum point contact in an integer quantum Hall (IQH) system. Shot-noise measurements reveal a clear transition from elementary-charge tunneling at low bias to fractional-charge tunneling at high bias. The fractional shot noise is proportional to T(1)(1-T(1)) over a wide range of T(1), where T(1) is the transmission probability of the IQH edge channel. This binomial distribution indicates that fractional quasiparticles emerge from the IQH state to be transmitted through the local FQH state. The study of this tunneling process enables us to elucidate the dynamics of Laughlin quasiparticles in FQH systems.

Recent experimental progress of fractional quantum Hall effect: 5/2 filling state and graphene

The phenomenon of fractional quantum Hall effect (FQHE) was first experimentally observed 33 years ago. FQHE involves strong Coulomb interactions and correlations among the electrons, which leads to quasiparticles with fractional elementary charge. Three decades later, the field of FQHE is still active with new discoveries and new technical developments. A significant portion of attention in FQHE has been dedicated to filling factor 5/2 state, for its unusual even denominator and possible application in topological quantum computation. Traditionally, FQHE has been observed in high-mobility GaAs heterostructure, but new materials such as graphene also open up a new area for FQHE. This review focuses on recent progress of FQHE at 5/2 state and FQHE in graphene.

Non-universality of scaling exponents in quantum Hall transitions

We have investigated experimentally the scaling behaviour of quantum Hall transitions in GaAs/AlGaAs heterostructures of a range of mobility, carrier concentration, and spacer layer width. All three critical scaling exponents γ, κ and p were determined independently for each sample. We measure the localization length exponent to be γ ≈ 2.3, in good agreement with expected predictions from scaling theory, but κ and p are found to possess non-universal values. Results obtained for κ range from κ = 0.16 ± 0.02 to κ = 0.67 ± 0.02, and are found to be Landau level (LL) dependent, whereas p is found to decrease with increasing sample mobility. Our results demonstrate the existence of two transport regimes in the LL conductivity peak; universality is found within the quantum coherent transport regime present in the tails of the conductivity peak, but is absent within the classical transport regime found close to the critical point at the centre of the conductivity peak. We explain these results using a percolation model and show that the critical scaling exponent depends on certain important length scales that correspond to the microscopic description of electron transport in the bulk of a two-dimensional electron system.

Transport in disordered monolayer MoS2 nanoflakes--evidence for inhomogeneous charge transport

We study charge transport in a monolayer MoS2 nanoflake over a wide range of carrier density, temperature and electric bias. We find that the transport is best described by a percolating picture in which the disorder breaks translational invariance, breaking the system up into a series of puddles, rather than previous pictures in which the disorder is treated as homogeneous and uniform. Our work provides insight to a unified picture of charge transport in monolayer MoS2 nanoflakes and contributes to the development of next-generation MoS2-based devices.

Signatures for Wigner crystal formation in the chemical potential of a two-dimensional electron system

We investigate the evolution of the chemical potential of a two-dimensional electron system (2DES) as a function of density at a fixed magnetic field. By using a bilayer system, changes in the chemical potential of one 2DES are determined from the density variation induced in the second, nearby 2DES. At high magnetic fields around a filling factor of ν=1 or ν=2, the chemical potential jump associated with the condensation in a quantum Hall state exhibits two anomalies symmetrically located around these integer filling factors. They are attributed to the formation of a 2D Wigner crystal of quasiparticles.

Local electrostatic imaging of striped domain order in LaAlO3/SrTiO3

The emerging field of complex oxide interfaces is generically built on one of the most celebrated substrates--strontium titanate (SrTiO3). This material hosts a range of phenomena, including ferroelasticity, incipient ferroelectricity, and most puzzlingly, contested giant piezoelectricity. Although these properties may markedly influence the oxide interfaces, especially on microscopic length scales, the lack of local probes capable of studying such buried systems has left their effects largely unexplored. Here we use a scanning charge detector--a nanotube single-electron transistor--to non-invasively image the electrostatic landscape and local mechanical response in the prototypical LaAlO3/SrTiO3 system with unprecedented sensitivity. Our measurements reveal that on microscopic scales SrTiO3 exhibits large anomalous piezoelectricity with curious spatial dependence. Through electrostatic imaging we unravel the microscopic origin for this extrinsic piezoelectricity, demonstrating its direct, quantitative connection to the motion of locally ordered tetragonal domains under applied gate voltage. These domains create striped potential modulations that can markedly influence the two-dimensional electron system at the conducting interface. Our results have broad implications to all complex oxide interfaces built on SrTiO3 and demonstrate the importance of microscopic structure to the physics of electrons at the LaAlO3/SrTiO3 interface.

Tunable insulator-quantum Hall transition in a weakly interacting two-dimensional electron system

We have performed low-temperature measurements on a gated two-dimensional electron system in which electron-electron (e-e) interactions are insignificant. At low magnetic fields, disorder-driven movement of the crossing of longitudinal and Hall resistivities (ρxx and ρxy) can be observed. Interestingly, by applying different gate voltages, we demonstrate that such a crossing at ρxx ~ ρxy can occur at a magnetic field higher, lower, or equal to the temperature-independent point in ρxx which corresponds to the direct insulator-quantum Hall transition. We explicitly show that ρxx ~ ρxy occurs at the inverse of the classical Drude mobility 1/μD rather than the crossing field corresponding to the insulator-quantum Hall transition. Moreover, we show that the background magnetoresistance can affect the transport properties of our device significantly. Thus, we suggest that great care must be taken when calculating the renormalized mobility caused by e-e interactions.

Coherent tunnelling across a quantum point contact in the quantum Hall regime

The unique properties of quantum hall devices arise from the ideal one-dimensional edge states that form in a two-dimensional electron system at high magnetic field. Tunnelling between edge states across a quantum point contact (QPC) has already revealed rich physics, like fractionally charged excitations, or chiral Luttinger liquid. Thanks to scanning gate microscopy, we show that a single QPC can turn into an interferometer for specific potential landscapes. Spectroscopy, magnetic field and temperature dependences of electron transport reveal a quantitatively consistent interferometric behavior of the studied QPC. To explain this unexpected behavior, we put forward a new model which relies on the presence of a quantum Hall island at the centre of the constriction as well as on different tunnelling paths surrounding the island, thereby creating a new type of interferometer. This work sets the ground for new device concepts based on coherent tunnelling.

Detecting non-Abelian anyons by charging spectroscopy

Observation of non-Abelian statistics for the e/4 quasiparticles in the ν = 5/2 fractional quantum Hall state remains an outstanding experimental problem. The non-Abelian statistics are linked to the presence of additional low energy states in a system with localized quasiparticles, and, hence, an additional low temperature entropy. Recent experiments, which detect changes in the number of quasiparticles trapped in a local potential well as a function of an applied gate voltage, V(G), provide a possibility for measuring this entropy, if carried out over a suitable range of temperatures, T. We present a microscopic model for quasiparticles in a potential well and study the effects of non-Abelian statistics on the charge stability diagram in the V(G)-T plane, including broadening at finite temperature. We predict a measurable slope for the first quasiparticle charging line and an even-odd effect in the diagram, which is a signature of non-Abelian statistics.

Unraveling quantum Hall breakdown in bilayer graphene with scanning gate microscopy

Investigating the structure of quantized plateaus in the Hall conductance of graphene is a powerful way of probing its crystalline and electronic structure and will also help to establish whether graphene can be used as a robust standard of resistance for quantum metrology. We use low-temperature scanning gate microscopy to image the interplateau breakdown of the quantum Hall effect in an exfoliated bilayer graphene flake. Scanning gate images captured during breakdown exhibit intricate patterns where the conductance is strongly affected by the presence of the scanning probe tip. The maximum density and intensity of the tip-induced conductance perturbations occur at half-integer filling factors, midway between consecutive quantum Hall plateau, while the intensity of individual sites shows a strong dependence on tip-voltage. Our results are well-described by a model based on quantum percolation which relates the points of high responsivity to tip-induced scattering in a network of saddle points separating localized states.

Transconductance fluctuations as a probe for interaction-induced quantum Hall states in graphene

Transport measurements normally provide a macroscopic, averaged view of the sample so that disorder prevents the observation of fragile interaction-induced states. Here, we demonstrate that transconductance fluctuations in a graphene field effect transistor reflect charge localization phenomena on the nanometer scale due to the formation of a dot network which forms near incompressible quantum states. These fluctuations give access to fragile broken symmetry and fractional quantum Hall states even though these states remain hidden in conventional magnetotransport quantities.

Imaging of Coulomb-driven quantum Hall edge states

The edges of a two-dimensional electron gas (2DEG) in the quantum Hall effect (QHE) regime are divided into alternating metallic and insulating strips, with their widths determined by the energy gaps of the QHE states and the electrostatic Coulomb interaction. Local probing of these submicrometer features, however, is challenging due to the buried 2DEG structures. Using a newly developed microwave impedance microscope, we demonstrate the real-space conductivity mapping of the edge and bulk states. The sizes, positions, and field dependence of the edge strips around the sample perimeter agree quantitatively with the self-consistent electrostatic picture. The evolution of microwave images as a function of magnetic fields provides rich microscopic information around the ν=2 QHE state.

Local charge of the ν = 5/2 fractional quantum Hall state

Electrons moving in two dimensions under the influence of strong magnetic fields effectively lose their kinetic energy and display exotic behaviour dominated by Coulomb forces. When the ratio of electrons to magnetic flux quanta in the system (ν) is near 5/2, the electrons are predicted to condense into a correlated phase with fractionally charged quasiparticles and a ground-state degeneracy that grows exponentially as these quasiparticles are introduced. The only way for electrons to transform between the many ground states would be to braid the fractional excitations around each other. This property has been proposed as the basis of a fault-tolerant quantum computer. Here we present observations of localized quasiparticles at ν = 5/2, confined to puddles by disorder. Using a local electrometer to compare how quasiparticles at ν = 5/2 and ν = 7/3 charge these puddles, we were able to extract the ratio of local charges for these states. Averaged over several disorder configurations and samples, we found the ratio to be 4/3, suggesting that the local charges are = e/3 and = e/4, where e is the charge of an electron. This is in agreement with theoretical predictions for a paired state at ν = 5/2. Confirming the existence of localized e/4 quasiparticles shows that proposed interferometry experiments to test statistics and computational ability of the state at ν = 5/2 would be possible.

Invited review article: A 10 mK scanning probe microscopy facility

We describe the design, development and performance of a scanning probe microscopy (SPM) facility operating at a base temperature of 10 mK in magnetic fields up to 15 T. The microscope is cooled by a custom designed, fully ultra-high vacuum (UHV) compatible dilution refrigerator (DR) and is capable of in situ tip and sample exchange. Subpicometer stability at the tip-sample junction is achieved through three independent vibration isolation stages and careful design of the dilution refrigerator. The system can be connected to, or disconnected from, a network of interconnected auxiliary UHV chambers, which include growth chambers for metal and semiconductor samples, a field-ion microscope for tip characterization, and a fully independent additional quick access low temperature scanning tunneling microscope (STM) and atomic force microscope (AFM) system. To characterize the system, we present the cooling performance of the DR, vibrational, tunneling current, and tip-sample displacement noise measurements. In addition, we show the spectral resolution capabilities with tunneling spectroscopy results obtained on an epitaxial graphene sample resolving the quantum Landau levels in a magnetic field, including the sublevels corresponding to the lifting of the electron spin and valley degeneracies.

Metal to insulator transition on the N=0 Landau level in graphene

The magnetotransport in single layer graphene has been experimentally investigated in magnetic fields up to 18 T as a function of temperature. A pronounced T dependence is observed for T≲50  K, which is either metallic, or insulating, depending on the filling factor ν. The metal-insulator transition (MIT) occurs at |ν{c}|∼0.65 and in the regime of the dissipative transport, where the longitudinal resistance Rxx>1/2R{K}. The critical resistivity (Rxx per square) is ρ{xx}(ν{c})≈1/2R{K} and is correlated with the appearance of zero plateau in Hall conductivity σ{xy}(ν) and peaks in σ{xx}(ν). This leads us to construct a universal low-T (n, B) phase diagram of this quantum phase transition.

Observable bulk signatures of non-Abelian quantum Hall States

We show that non-Abelian quantum Hall states can be identified by experimental measurements of the temperature dependence of either the electrochemical potential or the orbital magnetization. The predicted signals of non-Abelian statistics are within experimental resolution, and can be clearly distinguished from other contributions under realistic circumstances. The proposed measurement technique also has the potential to resolve spin-ordering transitions in low density electronic systems in the Wigner crystal and strongly interacting Luttinger liquid regimes.

Low-temperature and high magnetic field dynamic scanning capacitance microscope

We demonstrate a dynamic scanning capacitance microscope (DSCM) that operates at large bandwidths, cryogenic temperatures, and high magnetic fields. The setup is based on a noncontact atomic force microscope (AFM) with a quartz tuning fork sensor for the nonoptical excitation and readout in topography, force, and dissipation measurements. The metallic AFM tip forms part of a rf resonator with a transmission characteristics modulated by the sample properties and the tip-sample capacitance. The tip motion gives rise to a modulation of the capacitance at the frequency of the AFM sensor and its harmonics, which can be recorded simultaneously with the AFM data. We use an intuitive model to describe and analyze the resonator transmission and show that for most experimental conditions it is proportional to the complex tip-sample conductance, which depends on both the tip-sample capacitance and the sample resistivity. We demonstrate the performance of the DSCM on metal disks buried under a polymer layer and we discuss images recorded on a two-dimensional electron gas in the quantum Hall effect regime, i.e. at cryogenic temperatures and in high magnetic fields, where we directly image the formation of compressible stripes at the physical edge of the sample.

Quantum Hall transition in real space: from localized to extended states

Using scanning tunneling spectroscopy in an ultrahigh vacuum at low temperature (T=0.3 K) and high magnetic fields (B<or=12 T), we directly probe electronic wave functions across an integer quantum Hall transition. In accordance with theoretical predictions, we observe the evolution from localized drift states in the insulating phases to branched extended drift states at the quantum critical point. The observed microscopic behavior close to the extended state indicates points of localized quantum tunneling, which are considered to be decisive for a quantitative description of the transition.

Charge rearrangement and screening in a quantum point contact

Compressibility measurements are performed on a quantum point contact (QPC). Screening due to mobile charges in the QPC is measured quantitatively, using a second point contact. These measurements are performed from pinch-off through the opening of the first few modes in the QPC. While the measured signal closely matches a Thomas-Fermi-Poisson prediction, deviations from the classical behavior are apparent near the openings of the different modes. Density functional calculations attribute the deviations to a combination of a diverging density of states at the opening of each one-dimensional mode and exchange interaction, which is strongest for the first mode.

Influence of interactions on flux and back-gate period of quantum Hall interferometers

In quantum Hall systems with two narrow constrictions, tunneling between opposite edges can give rise to quantum interference and Aharonov-Bohm-like oscillations of the conductance. When there is an integer quantized Hall state within the constrictions, a region between them, with higher electron density, may form a compressible island. Electron tunneling through this island can lead to residual transport, modulated by Coulomb-blockade-type effects. We find that the coupling between the fully occupied lower Landau levels and the higher-partially occupied level gives rise to flux subperiods smaller than one flux quantum. We generalize this scenario to other geometries and to fractional quantum Hall systems, and compare our predictions to experiments.

Real-space imaging of alternate localization and extension of quasi-two-dimensional electronic States at graphite surfaces in magnetic fields

We measured the local density of states (LDOS) of a quasi-two-dimensional (2D) electron system near point defects on a surface of highly oriented pyrolytic graphite with scanning tunneling microscopy and spectroscopy. Differential tunnel conductance images taken at very low temperatures and in high magnetic fields show a clear contrast between localized and extended spatial distributions of the LDOS at the valley and peak energies of the Landau level spectrum, respectively. The localized electronic state has a single circular distribution around the defects with a radius comparable to the magnetic length. The localized LDOS is in good agreement with a spatial distribution of a calculated wave function for a single electron in 2D in a Coulomb potential in magnetic fields.

Resonant rayleigh scattering from bilayer quantum Hall phases

We observe resonant Rayleigh scattering of light from quantum Hall bilayers at Landau level filling factor nu = 1. The effect arises below 1 Kelvin when electrons are in the incompressible quantum Hall phase with strong interlayer correlations. Marked changes in the Rayleigh scattering signal in response to application of an in-plane magnetic field indicate that the unexpected temperature dependence is linked to formation of a nonuniform electron fluid close to the phase transition towards the compressible state. These results demonstrate a new realm of study in which resonant Rayleigh scattering methods probe quantum phases of electrons in semiconductor heterostructures.

Electron correlations and single-particle physics in the integer quantum Hall effect

The compressibility of a two-dimensional electron system with spin in a spatially correlated random potential and a quantizing magnetic field is investigated. Electron-electron interaction is treated with the Hartree-Fock method. Numerical results for the influences of interaction and disorder on the compressibility as a function of the particle density and the strength of the magnetic field are presented. Localization-delocalization transitions associated with a highly compressible region in the energy spectrum are found at half-integer filling factors. Coulomb blockade effects are found near integer fillings in the regions of low compressibility. Results are compared with recent experiments.

A single electron transistor on an atomic force microscope probe

We report fabrication as well as proof-of-concept experiments of a noninvasive sensor of weak nanoscale electric fields. The sensor is a single electron transistor (SET) placed at the tip of a noncontact atomic force microscope (AFM). This is a general technology to make any nanometer-sized lithography pattern at edges or tips of a cantilever. The height control of the AFM allows the SET to hover a few nanometers above the substrate, improving both the electric field sensitivity and lateral resolution of the electrometer. Our AFM-SET sensor is prepared by a scalable technology. It means that the probe can be routinely fabricated and replaced, if broken.

Imaging transport resonances in the quantum Hall effect

We use a scanning capacitance probe to image transport in the quantum Hall system. Applying a dc bias voltage to the tip induces a ring-shaped incompressible strip (IS) in the 2D electron system (2DES) that moves with the tip. At certain tip positions, short-range disorder in the 2DES creates a quantum dot island in the IS. These islands enable resonant tunneling across the IS, enhancing its conductance by more than 4 orders of magnitude. The images provide a quantitative measure of disorder and suggest resonant tunneling as the primary mechanism for transport across ISs.

Terahertz emission from quantum cascade lasers in the quantum Hall regime: evidence for many body resonances and localization effects

A terahertz quantum cascade laser, operating at lambda=159 microm and exploiting the in-plane confinement arising from perpendicular magnetic field, is used to investigate the physics of electrons confined on excited subbands in the regime of a large ratio of the magnetic field confinement energy to the photon energy. As the magnetic field is increased above about 6 T, and the temperature lowered below 20 K, the devices are characterized by a very low threshold current density, with values as low as J(th)=1A/cm(2), and an increase of gain by five times the low field value. We show that, as with the quantum Hall effect, the key physical process is the localization of the carriers. Evidences for resonant electron-electron scattering processes are directly obtained from light intensity and transport measurements.

Spatially resolved manipulation of single electrons in quantum dots using a scanned probe

The scanning metallic tip of a scanning force microscope was coupled capacitively to electrons confined in a lithographically defined gate-tunable quantum dot at a temperature of 300 mK. Single electrons were made to hop on or off the dot by moving the tip or by changing the tip bias voltage owing to the Coulomb-blockade effect. Spatial images of conductance resonances map the interaction potential between the tip and individual electronic quantum dot states. Under certain conditions this interaction is found to contain a tip-voltage induced and a tip-voltage-independent contribution.

Localization of fractionally charged quasi-particles

An outstanding question pertaining to the microscopic properties of the fractional quantum Hall effect is understanding the nature of the particles that participate in the localization but that do not contribute to electronic transport. By using a scanning single electron transistor, we imaged the individual localized states in the fractional quantum Hall regime and determined the charge of the localizing particles. Highlighting the symmetry between filling factors 1/3 and 2/3, our measurements show that quasi-particles with fractional charge e* = e/3 localize in space to submicrometer dimensions, where e is the electron charge.