Subsurface charge accumulation imaging of a quantum Hall liquid

Mapping charge excitations in generalized Wigner crystals

Transition metal dichalcogenide-based moiré superlattices exhibit strong electron-electron correlations, thus giving rise to strongly correlated quantum phenomena such as generalized Wigner crystal states. Evidence of Wigner crystals in transition metal dichalcogenide moire superlattices has been widely reported from various optical spectroscopy and electrical conductivity measurements, while their microscopic nature has been limited to the basic lattice structure. Theoretical studies predict that unusual quasiparticle excitations across the correlated gap between upper and lower Hubbard bands can arise due to long-range Coulomb interactions in generalized Wigner crystal states. However, the microscopic proof of such quasiparticle excitations is challenging because of the low excitation energy of the Wigner crystal. Here we describe a scanning single-electron charging spectroscopy technique with nanometre spatial resolution and single-electron charge resolution that enables us to directly image electron and hole wavefunctions and to determine the thermodynamic gap of generalized Wigner crystal states in twisted WS2 moiré heterostructures. High-resolution scanning single-electron charging spectroscopy combines scanning tunnelling microscopy with a monolayer graphene sensing layer, thus enabling the generation of individual electron and hole quasiparticles in generalized Wigner crystals. We show that electron and hole quasiparticles have complementary wavefunction distributions and that thermodynamic gaps of ∼50 meV exist for the 1/3 and 2/3 generalized Wigner crystal states in twisted WS2.

Edge-State Wave Functions from Momentum-Conserving Tunneling Spectroscopy

We perform momentum-conserving tunneling spectroscopy using a GaAs cleaved-edge overgrowth quantum wire to investigate adjacent quantum Hall edge states. We use the lowest five wire modes with their distinct wave functions to probe each edge state and apply magnetic fields to modify the wave functions and their overlap. This reveals an intricate and rich tunneling conductance fan structure which is succinctly different for each of the wire modes. We self-consistently solve the Poisson-Schrödinger equations to simulate the spectroscopy, reproducing the striking fans in great detail, thus, confirming the calculations. Further, the model predicts hybridization between wire states and Landau levels, which is also confirmed experimentally. This establishes momentum-conserving tunneling spectroscopy as a powerful technique to probe edge state wave functions.

Imaging Andreev Reflection in Graphene

Coherent charge transport along ballistic paths can be introduced into graphene by Andreev reflection, for which an electron reflects from a superconducting contact as a hole, while a Cooper pair is transmitted. We use liquid-helium cooled scanning gate microscopy (SGM) to image Andreev reflection in graphene in the magnetic focusing regime, where carriers move along cyclotron orbits between contacts. Images of flow are obtained by deflecting carrier paths and displaying the resulting change in conductance. When electrons enter the superconductor, Andreev-reflected holes leave for the collecting contact. To test the results, we destroy Andreev reflection with a large current and by heating above the critical temperature. In both cases, the reflected carriers change from holes to electrons.

Scanning nuclear resonance imaging of a hyperfine-coupled quantum Hall system

Nuclear resonance (NR) is widely used to detect and characterise nuclear spin polarisation and conduction electron spin polarisation coupled by a hyperfine interaction. While the macroscopic aspects of such hyperfine-coupled systems have been addressed in most relevant studies, the essential role of local variation in both types of spin polarisation has been indicated in 2D semiconductor systems. In this study, we apply a recently developed local and highly sensitive NR based on a scanning probe to a hyperfine-coupled quantum Hall (QH) system in a 2D electron gas subject to a strong magnetic field. We succeed in imaging the NR intensity and Knight shift, uncovering the spatial distribution of both the nuclear and electron spin polarisation. The results reveal the microscopic origin of the nonequilibrium QH phenomena, and highlight the potential use of our technique in microscopic studies on various electron spin systems as well as their correlations with nuclear spins.

An incompressible state of a photo-excited electron gas

Two-dimensional electrons in a magnetic field can form new states of matter characterized by topological properties and strong electronic correlations as displayed in the integer and fractional quantum Hall states. In these states, the electron liquid displays several spectacular characteristics, which manifest themselves in transport experiments with the quantization of the Hall resistance and a vanishing longitudinal conductivity or in thermodynamic equilibrium when the electron fluid becomes incompressible. Several experiments have reported that dissipationless transport can be achieved even at weak, non-quantizing magnetic fields when the electrons absorb photons at specific energies related to their cyclotron frequency. Here we perform compressibility measurements on electrons on liquid helium demonstrating the formation of an incompressible electronic state under these resonant excitation conditions. This new state provides a striking example of irradiation-induced self-organization in a quantum system.

Two-dimensional phases of electrons exhibit interesting phenomena under magnetic fields. Chepelianskii et al. show that electrons on liquid helium exhibit an incompressible state when they are excited by a microwave field at particular frequencies related with the Landau level spacing.

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.

Counting elementary charges on nanoparticles by electron holography

The distribution and movement of charge is fundamental to many physical phenomena, particularly for applications involving nanoparticles, nanostructures, and electronic devices. However, there are very few ways of quantifying charge at the necessary length scale. Here, we show that aberration-corrected electron holography is capable of counting the charge on individual nanoparticles to a precision of one elementary unit of charge. We present a method that measures charges within predefined contours by directly applying Gauss's law at the nanoscale. We perform a statistical analysis to reveal the relationship between the size of the contours and the precision of the charge measurement and present strategies to optimize the spatial and signal resolution for the presented method.

Design of a scanning gate microscope for mesoscopic electron systems in a cryogen-free dilution refrigerator

We report on our design of a scanning gate microscope housed in a cryogen-free dilution refrigerator with a base temperature of 15 mK. The recent increase in efficiency of pulse tube cryocoolers has made cryogen-free systems popular in recent years. However, this new style of cryostat presents challenges for performing scanning probe measurements, mainly as a result of the vibrations introduced by the cryocooler. We demonstrate scanning with root-mean-square vibrations of 0.8 nm at 3 K and 2.1 nm at 15 mK in a 1 kHz bandwidth with our design. Using Coulomb blockade thermometry on a GaAs/AlGaAs gate-defined quantum dot, we demonstrate an electron temperature of 45 mK.

Imaging fractional incompressible stripes in integer quantum Hall systems

Transport experiments provide conflicting evidence on the possible existence of fractional order within integer quantum Hall systems. In fact, integer edge states sometimes behave as monolithic objects with no inner structure, while other experiments clearly highlight the role of fractional substructures. Recently developed low-temperature scanning probe techniques offer today an opportunity for a deeper-than-ever investigation of spatial features of such edge systems. Here we use scanning-gate microscopy and demonstrate that fractional features were unambiguously observed in every integer quantum Hall constriction studied. We present also an experimental estimate of the width of the fractional incompressible stripes corresponding to filling factors 1/3, 2/5, 3/5, and 2/3. Our results compare well with predictions of the edge-reconstruction theory.

Near-field optical mapping of quantum Hall edge states

We report on the mapping of quantum-Hall edge states by quasiresonant photovoltage measurements using a near-field scanning optical microscope. We have observed fine structures near sample edges that shift inward with an increase in magnetic field in accordance with the shift of the positions of the quantum-Hall edge states. We have found a transition from the weak disorder regime where compressible-incompressble strips are visible to the strong disorder regime where fluctuations smear out incompressible strips.

Single-electron capacitance spectroscopy of individual dopants in silicon

Motivated by recent transport experiments and proposed atomic-scale semiconductor devices, we present measurements that extend the reach of scanned-probe methods to discern the properties of individual dopants tens of nanometers below the surface of a silicon sample. Using a capacitance-based approach, we have both spatially resolved individual subsurface boron acceptors and detected spectroscopically single holes entering and leaving these minute systems of atoms. A resonance identified as the B+ state is shown to shift in energy from acceptor to acceptor. We examine this behavior with respect to nearest-neighbor distances. By directly measuring the quantum levels and testing the effect of dopant-dopant interactions, this method represents a valuable tool for the development of future atomic-scale semiconductor devices.

Scanning gate imaging of quantum dots in 1D ultra-thin InAs/InP nanowires

We use a scanning gate microscope (SGM) to characterize one-dimensional ultra-thin (diameter ≈ 30 nm) InAs/InP heterostructure nanowires containing a nominally 300 nm long InAs quantum dot defined by two InP tunnel barriers. Measurements of Coulomb blockade conductance versus backgate voltage with no tip present are difficult to decipher. Using the SGM tip as a charged movable gate, we are able to identify three quantum dots along the nanowire: the grown-in quantum dot and an additional quantum dot near each metal lead. The SGM conductance images are used to disentangle information about individual quantum dots and then to characterize each quantum dot using spatially resolved energy-level spectroscopy.

Imaging Coulomb islands in a quantum Hall interferometer

In the quantum Hall regime, near integer filling factors, electrons should only be transmitted through spatially separated edge states. However, in mesoscopic systems, electronic transmission turns out to be more complex, giving rise to a large spectrum of magnetoresistance oscillations. To explain these observations, recent models put forward the theory that, as edge states come close to each other, electrons can hop between counterpropagating edge channels, or tunnel through Coulomb islands. Here, we use scanning gate microscopy to demonstrate the presence of QH Coulomb islands, and reveal the spatial structure of transport inside a QH interferometer. Locations of electron islands are found by modulating the tunnelling between edge states and confined electron orbits. Tuning the magnetic field, we unveil a continuous evolution of active electron islands. This allows to decrypt the complexity of high-magnetic-field magnetoresistance oscillations, and opens the way to further local-scale manipulations of QH localized states.

Charge Writing at the LaAlO3/SrTiO3 surface

Biased conducting-tip atomic force microscopy (AFM) has been shown to write and erase nanoscale metallic lines at the LaAlO(3)/SrTiO(3) interface. Using various AFM modes, we show the mechanism of conductivity switching is the writing of surface charge. These charges are stably deposited on a wide range of LaAlO(3) thicknesses, including bulk crystals. A strong asymmetry with writing polarity was found for 1 and 2 unit cells of LaAlO(3), providing experimental evidence for a theoretically predicted built-in potential.

Imaging coherent transport in graphene. Part I: mapping universal conductance fluctuations

Graphene provides a fascinating testbed for new physics and exciting opportunities for future applications based on quantum phenomena. To understand the coherent flow of electrons through a graphene device, we employ a nanoscale probe that can access the relevant length scales--the tip of a liquid-He-cooled scanning probe microscope (SPM) capacitively couples to the graphene device below, creating a movable scatterer for electron waves. At sufficiently low temperatures and small size scales, the diffusive transport of electrons through graphene becomes coherent, leading to universal conductance fluctuations (UCF). By scanning the tip over a device, we map these conductance fluctuations versus scatterer position. We find that the conductance is highly sensitive to the tip position, producing delta G approximately e(2)/h fluctuations when the tip is displaced by a distance comparable to half the Fermi wavelength. These measurements are in good agreement with detailed quantum simulations of the imaging experiment and demonstrate the value of a cooled SPM for probing coherent transport in graphene.

Imaging coherent transport in graphene. Part II: probing weak localization

Graphene has opened new avenues of research in quantum transport, with potential applications for coherent electronics. Coherent transport depends sensitively on scattering from microscopic disorder present in graphene samples: electron waves traveling along different paths interfere, changing the total conductance. Weak localization is produced by the coherent backscattering of waves, while universal conductance fluctuations are created by summing over all paths. In this work, we obtain conductance images of weak localization with a liquid-He-cooled scanning probe microscope, by using the tip to create a movable scatterer in a graphene device. This technique allows us to investigate coherent transport with a probe of size comparable to the electron wavelength. Images of magnetoconductance versus tip position map the effects of disorder by moving a single scatterer, revealing how electron interference is modified by the tip perturbation. The weak localization dip in conductivity at B = 0 is obtained by averaging magnetoconductance traces at different positions of the tip-created scatterer. The width Delta B(WL) of the dip yields an estimate of the electron coherence length L(phi) at fixed charge density. This 'scanning scatterer' method provides a new way of investigating coherent transport in graphene by directly perturbing the disorder configuration that creates these interferometric effects.

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.

Modeling single- and multiple-electron resonances for electric-field-sensitive scanning probes

We have developed a modeling method suitable for analyzing single- and multiple-electron resonances detected by electric-field-sensitive scanning probe techniques. The method is based on basic electrostatics and a numerical boundary-element approach. The results compare well to approximate analytical expressions and experimental data.

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.

Imaging electron interferometer

An imaging interferometer was created in a two-dimensional electron gas by reflecting electron waves emitted from a quantum point contact with a circular mirror. Images of electron flow obtained with a scanning probe microscope at liquid He temperatures show interference fringes when the mirror is energized. A quantum phase shifter was created by moving the mirror via its gate voltage, and an interferometric spectrometer can be formed by sweeping the tip over many wavelengths. Experiments and theory demonstrate that the interference signal is robust against thermal averaging.

Modeling electric-field-sensitive scanning probe measurements for a tip of arbitrary shape

We present a numerical method to model electric-field-sensitive scanning probe microscopy measurements which allows for a tip of arbitrary shape and invokes image charges to exactly account for a sample dielectric overlayer. The method is applied to calculate the spatial resolution of a subsurface charge accumulation imaging system, achieving reasonable agreement with experiment.

Imaging of cyclotron emission from edge channels in quantum Hall conductors

A local probing technique of cyclotron emission is applied to image nonequilibrium electrons generated along edge channels in quantum Hall conductors. In a lower-magnetic field region of a quantum Hall state plateau (filling factor 2<nu), cyclotron emission is found to occur along the boundary of a conductor on the side of lower potential (with positive Hall voltage). The emission indicates that electrons are accumulated in the first-excited Landau level, while they are depleted in the lowest Landau level along the boundary as a consequence of adiabatic-transport.

Erasable electrostatic lithography for quantum components

Quantum electronic components--such as quantum antidots and one-dimensional channels--are usually defined from doped GaAs/AlGaAs heterostructures using electron-beam lithography or local oxidation by conductive atomic force microscopy. In both cases, lithography and measurement are performed in very different environments, so fabrication and test cycles can take several weeks. Here we describe a different lithographic technique, which we call erasable electrostatic lithography (EEL), where patterns of charge are drawn on the device surface with a negatively biased scanning probe in the same low-temperature high-vacuum environment used for measurement. The charge patterns locally deplete electrons from a subsurface two-dimensional electron system (2DES) to define working quantum components. Charge patterns are erased locally with the scanning probe biased positive or globally by illuminating the device with red light. We demonstrate and investigate EEL by drawing and erasing quantum antidots, then develop the technique to draw and tune high-quality one-dimensional channels. The quantum components are imaged using scanned gate microscopy. A technique similar to EEL has been reported previously, where tip-induced charging of the surface or donor layer was used to locally perturb a 2DES before charge accumulation imaging.

Direct comparison between potential landscape and local density of states in a disordered two-dimensional electron system

The local density of states (LDOS) of the adsorbate-induced two-dimensional electron system (2DES) on n-InAs(110) is studied by scanning tunneling spectroscopy. In contrast to a similar 3DES, the 2DES LDOS exhibits 20 times stronger corrugations and rather irregular structures. Both results are interpreted as consequences of weak localization. Fourier transforms of the LDOS reveal that the k values of the unperturbed 2DES still dominate the 2DES, but additional lower k values contribute. To clarify the origin of the LDOS patterns, we measure the potential landscape of the 2DES area. We use it to calculate the expected LDOS and find reasonable agreement between calculation and experiment.

Scanned probe imaging of single-electron charge states in nanotube quantum dots

An atomic force microscope was used to study single-electron motion in nanotube quantum dots. By applying a voltage to the microscope tip, the number of electrons occupying the quantum dot could be changed, causing Coulomb oscillations in the nanotube conductance. Spatial maps of these oscillations were used to locate individual dots and to study the electrostatic coupling between the dot and the tip. The electrostatic forces associated with single electrons hopping on and off the quantum dot were also measured. These forces changed the amplitude, frequency, and quality factor of the cantilever oscillation, demonstrating how single-electron motion can interact with a mechanical oscillator.

Topographic mapping of the quantum hall liquid using a few-electron bubble

A scanning probe technique was used to obtain a high-resolution map of the random electrostatic potential inside the quantum Hall liquid. A sharp metal tip, scanned above a semiconductor surface, sensed charges in an embedded two-dimensional (2D) electron gas. Under quantum Hall effect conditions, applying a positive voltage to the tip locally enhanced the 2D electron density and created a "bubble" of electrons in an otherwise unoccupied Landau level. As the tip scanned along the sample surface, the bubble followed underneath. The tip sensed the motions of single electrons entering or leaving the bubble in response to changes in the local 2D electrostatic potential.

Imaging of localized electronic states in the quantum Hall regime

The concept of electron localization has long been accepted to be essential to the physics of the quantum Hall effect in a two-dimensional electron gas. The exact quantization of the Hall resistance and the zero of the diagonal resistance over a range of filling factors close to integral are attributed to the localization of electronic states at the Fermi level in the interior of the gas. As the electron density is changed, charging of the individual localized states may occur by single-electron jumps, causing associated oscillations in the local electrostatic potential. Here we search for such a manifestation of localized states in the quantum Hall regime, using a scanning electrometer probe. We observe localized potential signals, at numerous locations, that oscillate with changing electron density. In general, the corresponding spatial patterns are complex, but well-defined objects are often seen which evidently arise from individual localized states. These objects interact, and at times form a lattice-like arrangement.

Tunnelling between the edges of two lateral quantum Hall systems

The edge of a two-dimensional electron system in a magnetic field consists of one-dimensional channels that arise from the confining electric field at the edge of the system. The crossed electric and magnetic fields cause electrons to drift parallel to the sample boundary, creating a chiral current that travels along the edge in only one direction. In an ideal two-dimensional electron system in the quantum Hall regime, all the current flows along the edge. Quantization of the Hall resistance arises from occupation of N one-dimensional edge channels, each contributing a conductance of e2/h. Here we report differential conductance measurements, in the integer quantum Hall regime, of tunnelling between the edges of a pair of two-dimensional electron systems that are separated by an atomically precise, high-quality, tunnel barrier. The resultant interaction between the edge states leads to the formation of new energy gaps and an intriguing dispersion relation for electrons travelling along the barrier: for example, we see a persistent conductance peak at zero bias voltage and an absence of tunnelling features due to electron spin. These features are unexpected and are not consistent with a model of weakly interacting edge states. Remnant disorder along the barrier and charge screening may each play a role, although detailed numerical studies will be required to elucidate these effects.