Coulomb barrier to tunneling between parallel two-dimensional electron systems

Spectroscopy of the fractal Hofstadter energy spectrum

Hofstadter's butterfly, the predicted energy spectrum for non-interacting electrons confined to a two-dimensional lattice in a magnetic field, is one of the most remarkable fractal structures in nature1. At rational ratios of magnetic flux quanta per lattice unit cell, this spectrum shows self-similar distributions of energy levels that reflect its recursive construction. For most materials, Hofstadter's butterfly is predicted under experimental conditions that are unachievable using laboratory-scale magnetic fields1-3. More recently, electrical transport studies have provided evidence for Hofstadter's butterfly in materials engineered to have artificially large lattice constants4-6, such as those with moiré superlattices7-10. Yet, so far, direct spectroscopy of the fractal energy spectrum predicted by Hofstadter nearly 50 years ago has remained out of reach. Here we use high-resolution scanning tunnelling microscopy/spectroscopy (STM/STS) to investigate the flat electronic bands in twisted bilayer graphene (TBG) near the predicted second magic angle11,12, an ideal setting for spectroscopic studies of Hofstadter's spectrum. Our study shows the fractionalization of flat moiré bands into discrete Hofstadter subbands and discerns experimental signatures of self-similarity of this spectrum. Moreover, our measurements uncover a spectrum that evolves dynamically with electron density, showing phenomena beyond that of Hofstadter's original model owing to the combined effects of strong correlations, Coulomb interactions and the quantum degeneracy of electrons in TBG.

Trion sensing of a zero-field composite Fermi liquid

The half-filled lowest Landau level is a fascinating platform for researching interacting topological phases. A celebrated example is the composite Fermi liquid, a non-Fermi liquid formed by composite fermions in strong magnetic fields1-10. Its zero-field counterpart is predicted in a twisted MoTe2 bilayer (tMoTe2)11,12-a recently discovered fractional Chern insulator exhibiting the fractional quantum anomalous Hall effect13-16. Although transport measurements at ν = -1/2 show signatures consistent with a zero-field composite Fermi liquid14, new probes are crucial to investigate the state and its elementary excitations. Here, by using the unique valley properties of tMoTe2, we report optical signatures of a zero-field composite Fermi liquid. We measured the degree of circular polarization (ρ) of trion photoluminescence versus hole doping and electric field. We found that, within the phase space showing robust ferromagnetism, ρ is near unity for Fermi liquid states. However, ρ is quenched at both integer and fractional Chern insulators, and in a hole doping range near ν = -1/2. Temperature, optical excitation power and electric-field-dependence measurements demonstrate that the quenching of ρ is a direct consequence of an energy gap (pseudogap) for electronic excitations of the Chern insulators (composite Fermi liquid): because the local spin-polarized excitations necessary to form trions are strongly suppressed, trion formation at the corresponding filling factors relies on optically generated unpolarized itinerant holes. Our work highlights a new excitonic probe of zero-field fractional Chern insulator physics, unique to tMoTe2.

Non-KAM classical chaos topology for electrons in superlattice minibands determines the inter-well quantum transition rates

We investigate the quantum-classical correspondence for a particle tunnelling through a periodic superlattice structure with an applied bias voltage and an additional tilted harmonic oscillator potential. We show that the quantum mechanical tunnelling rate between neighbouring quantum wells of the superlattice is determined by the topology of the phase trajectories of the analogous classical system. This result also enables us to estimate, with high accuracy, the tunnelling rate between two spatially displaced simple harmonic oscillator states using a classical model, and thus gain new insight into this generic quantum phenomenon. This finding opens new directions for exploring and understanding the quantum-classical correspondence principle and quantum jumps between displaced harmonic oscillators, which are important in many branches of natural science.

Composite Fermi Liquid at Zero Magnetic Field in Twisted MoTe_{2}

The pursuit of exotic phases of matter outside of the extreme conditions of a quantizing magnetic field is a long-standing quest of solid state physics. Recent experiments have observed spontaneous valley polarization and fractional Chern insulators in zero magnetic field in twisted bilayers of MoTe_{2}, at partial filling of the topological valence band (ν=-2/3 and -3/5). We study the topological valence band at half filling, using exact diagonalization and density matrix renormalization group calculations. We discover a composite Fermi liquid (CFL) phase even at zero magnetic field that covers a large portion of the phase diagram near twist angle ∼3.6°. The CFL is a non-Fermi liquid phase with metallic behavior despite the absence of Landau quasiparticles. We discuss experimental implications including the competition between the CFL and a Fermi liquid, which can be tuned with a displacement field. The topological valence band has excellent quantum geometry over a wide range of twist angles and a small bandwidth that is, remarkably, reduced by interactions. These key properties stabilize the exotic zero field quantum Hall phases. Finally, we present an optical signature involving "extinguished" optical responses that detects Chern bands with ideal quantum geometry.

Emergence of Interlayer Coherence in Twist-Controlled Graphene Double Layers

We report enhanced interlayer tunneling with reduced linewidth at zero interlayer bias in a twist-controlled double monolayer graphene heterostructure in the quantum Hall regime, when the top (ν_{T}) and bottom (ν_{B}) layer filling factors are near ν_{T}=±1/2,±3/2 and ν_{B}=±1/2,±3/2, and the total filling factor ν=±1 or ±3. The zero-bias interlayer conductance peaks are stable against variations of layer filling factor, and signal the emergence of interlayer phase coherence. Our results highlight twist control as a key attribute in revealing interlayer coherence using tunneling.

New disordered anyon phase of doped graphene zigzag nanoribbon

We investigate interacting disordered zigzag nanoribbons at low doping, using the Hubbard model to treat electron interactions within the density matrix renormalization group and Hartree-Fock method. Extra electrons that are inserted into an interacting disordered zigzag nanoribbon divide into anyons. Furthermore, the fractional charges form a new disordered anyon phase with a highly distorted edge spin density wave, containing numerous localized magnetic moments residing on the zigzag edges, thereby displaying spin-charge separation and a strong non-local correlation between the opposite zigzag edges. We make the following new predictions, which can be experimentally tested: (1) In the low doping case and weak disorder regime, the soft gap in the tunneling density of states is replaced by a sharp peak at the midgap energy with two accompanying peaks. The [Formula: see text] fractional charges that reside on the boundary of the zigzag edges are responsible for these peaks. (2) We find that the midgap peak disappears as the doping concentration increases. The presence of [Formula: see text] fractional charges will be strongly supported by the detection of these peaks. Doped zigzag ribbons may also exhibit unusual transport, magnetic, and inter-edge tunneling properties.

Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet

The interaction between electrons in graphene under high magnetic fields drives the formation of a rich set of quantum Hall ferromagnetic (QHFM) phases with broken spin or valley symmetry. Visualizing atomic-scale electronic wave functions with scanning tunneling spectroscopy (STS), we resolved microscopic signatures of valley ordering in QHFM phases and spectral features of fractional quantum Hall phases of graphene. At charge neutrality, we observed a field-tuned continuous quantum phase transition from a valley-polarized state to an intervalley coherent state, with a Kekulé distortion of its electronic density. Mapping the valley texture extracted from STS measurements of the Kekulé phase, we could visualize valley skyrmion excitations localized near charged defects. Our techniques can be applied to examine valley-ordered phases and their topological excitations in a wide range of materials.

Temperature-Induced Lifshitz Transition and Possible Excitonic Instability in ZrSiSe

The nodal-line semimetals have attracted immense interest due to the unique electronic structures such as the linear dispersion and the vanishing density of states as the Fermi energy approaching the nodes. Here, we report temperature-dependent transport and scanning tunneling microscopy (spectroscopy) [STM(S)] measurements on nodal-line semimetal ZrSiSe. Our experimental results and theoretical analyses consistently demonstrate that the temperature induces Lifshitz transitions at 80 and 106 K in ZrSiSe, which results in the transport anomalies at the same temperatures. More strikingly, we observe a V-shaped dip structure around Fermi energy from the STS spectrum at low temperature, which can be attributed to co-effect of the spin-orbit coupling and excitonic instability. Our observations indicate the correlation interaction may play an important role in ZrSiSe, which owns the quasi-two-dimensional electronic structures.

Resonant tunnelling spectroscopy of van der Waals heterosystems

The review concerns the most interesting aspects of (mainly experimental) resonance tunnelling spectroscopy studies of a new type of heterosystems called van der Waals heterostructures. The possibility to compose such systems is a result of the recent discovery of two-dimensional crystals, a new class of materials derived from graphene. The role of the angular mismatch of the crystal lattices of conductive graphene electrodes in the tunnelling of charge carriers between them, as well as the closely related issues associated with fulfillment of the conservation laws during tunnelling transitions are considered. The experimental results on inelastic tunnelling in the graphene/h-BN/graphene heterosystems with strong angular mismatch are discussed. The experiments made it possible to determine the phonon density of states spectra of the constituent layers and to detect and describe tunnelling transitions involving localized states of structural defects in the h-BN barrier. We consider new results of studies on tunnelling and magnetotunnelling in van der Waals heterosystems that demonstrate the possibilities of practical application of resonant tunnelling effects in, e.g., microwave engineering, based on realization of electronic devices having I – V curves with negative differential conductance (NDC) regions at tunnelling through defect levels of the barrier layers in such systems. These studies revealed two new types of heterosystems characterized by the formation of NDC regions as a result of resonant tunnelling through the defect levels in the h-BN barrier and by defect-assisted generation of tunnelling current.

The bibliography includes 40 references.

Precursors to Exciton Condensation in Quantum Hall Bilayers

Tunneling spectroscopy reveals evidence for interlayer electron-hole correlations in quantum Hall bilayer two-dimensional electron systems at layer separations near, but above, the transition to the incompressible exciton condensate at total Landau level filling ν_{T}=1. These correlations are manifested by a nonlinear suppression of the Coulomb pseudogap which inhibits low energy interlayer tunneling in weakly coupled bilayers. The pseudogap suppression is strongest at ν_{T}=1 and grows rapidly as the critical layer separation for exciton condensation is approached from above.

Effect of Magnetization on the Tunneling Anomaly in Compressible Quantum Hall States

Tunneling of electrons into a two-dimensional electron system is known to exhibit an anomaly at low bias, in which the tunneling conductance vanishes due to a many-body interaction effect. Recent experiments have measured this anomaly between two copies of the half-filled Landau level as a function of in-plane magnetic field, and they suggest that increasing spin polarization drives a deeper suppression of tunneling. Here, we present a theory of the tunneling anomaly between two copies of the partially spin-polarized Halperin-Lee-Read state, and we show that the conventional description of the tunneling anomaly, based on the Coulomb self-energy of the injected charge packet, is inconsistent with the experimental observation. We propose that the experiment is operating in a different regime, not previously considered, in which the charge-spreading action is determined by the compressibility of the composite fermions.

Quantum Hall Spin Diode

Double layer two-dimensional electron systems at high perpendicular magnetic field are used to realize magnetic tunnel junctions in which the electrons at the Fermi level in the two layers have either parallel or antiparallel spin magnetizations. In the antiparallel case the tunnel junction, at low temperatures, behaves as a nearly ideal spin diode. At elevated temperatures the diode character degrades as long-wavelength spin waves are thermally excited. These tunnel junctions provide a demonstration that the spin polarization of the electrons in the N=1 Landau level at filling factors ν=5/2 and 7/2 is essentially complete, and, with the aid of an in-plane magnetic field component, that Landau level mixing at these filling factors is weak in the samples studied.

Signatures of Phonon and Defect-Assisted Tunneling in Planar Metal-Hexagonal Boron Nitride-Graphene Junctions

Electron tunneling spectroscopy measurements on van der Waals heterostructures consisting of metal and graphene (or graphite) electrodes separated by atomically thin hexagonal boron nitride tunnel barriers are reported. The tunneling conductance, dI/dV, at low voltages is relatively weak, with a strong enhancement reproducibly observed to occur at around |V| ≈ 50 mV. While the weak tunneling at low energies is attributed to the absence of substantial overlap, in momentum space, of the metal and graphene Fermi surfaces, the enhancement at higher energies signals the onset of inelastic processes in which phonons in the heterostructure provide the momentum necessary to link the Fermi surfaces. Pronounced peaks in the second derivative of the tunnel current, d²I/dV², are observed at voltages where known phonon modes in the tunnel junction have a high density of states. In addition, features in the tunneling conductance attributed to single electron charging of nanometer-scale defects in the boron nitride are also observed in these devices. The small electronic density of states of graphene allows the charging spectra of these defect states to be electrostatically tuned, leading to "Coulomb diamonds" in the tunneling conductance.

Evidence for Defect-Mediated Tunneling in Hexagonal Boron Nitride-Based Junctions

We investigate electron tunneling through atomically thin layers of hexagonal boron nitride (hBN). Metal (Cr/Au) and semimetal (graphite) counter-electrodes are employed. While the direct tunneling resistance increases nearly exponentially with barrier thickness as expected, the thicker junctions also exhibit clear signatures of Coulomb blockade, including strong suppression of the tunnel current around zero bias and step-like features in the current at larger biases. The voltage separation of these steps suggests that single-electron charging of nanometer-scale defects in the hBN barrier layer are responsible for these signatures. We find that annealing the metal-hBN-metal junctions removes these defects and the Coulomb blockade signatures in the tunneling current.

Resonant tunnelling and negative differential conductance in graphene transistors

The chemical stability of graphene and other free-standing two-dimensional crystals means that they can be stacked in different combinations to produce a new class of functional materials, designed for specific device applications. Here we report resonant tunnelling of Dirac fermions through a boron nitride barrier, a few atomic layers thick, sandwiched between two graphene electrodes. The resonance occurs when the electronic spectra of the two electrodes are aligned. The resulting negative differential conductance in the device characteristics persists up to room temperature and is gate voltage-tuneable due to graphene’s unique Dirac-like spectrum. Although conventional resonant tunnelling devices comprising a quantum well sandwiched between two tunnel barriers are tens of nanometres thick, the tunnelling carriers in our devices cross only a few atomic layers, offering the prospect of ultra-fast transit times. This feature, combined with the multi-valued form of the device characteristics, has potential for applications in high-frequency and logic devices.

Multilayer stacks of graphene and related two-dimensional crystals can be tailored to create new classes of functional materials. Britnell et al. report resonant tunnelling of Dirac fermions and tunable negative differential conductance in a graphene-boron nitride-graphene transistor.

Theory of high-energy features in the tunneling spectra of quantum-Hall systems

We show that the low-temperature sash features in lowest Landau-level (LLL) tunneling spectra recently discovered by Dial and Ashoori are intimately related to the discrete Haldane-pseudopotential interaction energy scales that govern fractional quantum-Hall physics. Our analysis is based on expressions for the tunneling density of states which become exact at filling factors close to ν=0 and ν=1, where the sash structure is most prominent. We comment on other aspects of LLL correlation physics that can be revealed by accurate temperature-dependent tunneling data.

Evidence for a finite-temperature phase transition in a bilayer quantum Hall system

We study the Josephson-like interlayer tunneling signature of the strongly correlated nuT=1 quantum Hall phase in bilayer two-dimensional electron systems as a function of the layer separation, temperature, and interlayer charge imbalance. Our results offer strong evidence that a finite temperature phase transition separates the interlayer coherent phase from incoherent phases which lack strong interlayer correlations. The transition temperature is dependent on both the layer spacing and charge imbalance between the layers.

High-resolution spectroscopy of two-dimensional electron systems

Spectroscopic methods involving the sudden injection or ejection of electrons in materials are a powerful probe of electronic structure and interactions. These techniques, such as photoemission and tunnelling, yield measurements of the 'single-particle' density of states spectrum of a system. This density of states is proportional to the probability of successfully injecting or ejecting an electron in these experiments. It is equal to the number of electronic states in the system able to accept an injected electron as a function of its energy, and is among the most fundamental and directly calculable quantities in theories of highly interacting systems. However, the two-dimensional electron system (2DES), host to remarkable correlated electron states such as the fractional quantum Hall effect, has proved difficult to probe spectroscopically. Here we present an improved version of time-domain capacitance spectroscopy that allows us to measure the single-particle density of states of a 2DES with unprecedented fidelity and resolution. Using the method, we perform measurements of a cold 2DES, providing direct measurements of interesting correlated electronic effects at energies that are difficult to reach with other techniques; these effects include the single-particle exchange-enhanced spin gap, single-particle lifetimes in the quantum Hall system, and exchange splitting of Landau levels not at the Fermi surface.

Fractional charges and quantum phase transitions in imbalanced bilayer quantum Hall systems

We extend the composite boson theory to study slightly imbalanced bilayer quantum Hall systems. In the global U(1) symmetry breaking excitonic superfluid side, as the imbalance increases, the system supports continuously changing fractional charges. In the translational symmetry breaking pseudospin density wave (PSDW) side, there are two quantum phase transitions from the commensurate PSDW to an incommensurate PSDW and then to the excitonic superfluid state. We compare our theory with experimental data and also the previous microscopic calculations.

Reconstructing the electron in a fractionalized quantum fluid

The low-energy physics of the fractional Hall liquid is described in terms of quasiparticles that are qualitatively distinct from electrons. We show, however, that a long-lived electronlike quasiparticle also exists in the excitation spectrum: the state obtained by the application of an electron creation operator to a fractional quantum Hall ground state has a nonzero overlap with a complex, high energy bound state containing an odd number of composite-fermion quasiparticles. The electron annihilation operator similarly couples to a bound complex of composite-fermion holes. We predict that these bound states can be observed through a conductance resonance in experiments involving a tunneling of an external electron into the fractional quantum Hall liquid. A comment is made on the origin of the breakdown of the Fermi liquid paradigm in the fractional Hall liquid.

Spin transition in strongly correlated bilayer two-dimensional electron systems

Using a combination of heat pulse and nuclear magnetic resonance techniques, we demonstrate that the phase boundary separating the interlayer phase coherent quantum Hall effect at nu(T) = 1 in bilayer electron gases from the weakly coupled compressible phase depends upon the spin polarization of the nuclei in the host semiconductor crystal. Our results strongly suggest that, contrary to the usual assumption, the transition is attended by a change in the electronic spin polarization.

PSEUDO-GAP AND SPIN POLARIZATION IN A TWO-DIMENSIONAL ELECTRON GAS

Tunneling (one-particle excitation) density of states in the vicinity of Fermi level of a two-dimensional electron gas (2DEG) subjected to an external parallel and zero magnetic field is calculated. It reveals a pseudo-gap recently observed in the experiments. The gap originates in spin polarization of 2DEG. Nonmonotonic dependence of energy on a Landau level filling factor (density) for perpendicular magnetic field was also obtained. It implies the tunneling current peculiarities at filling factors of about 1/2 and 1. The Ising-like model of the exchange interaction in 2DEG was exploited instead of the conventional one. It was crucial to achieve even a qualitative agreement with experimental data.

Tunneling spectroscopy of a two-dimensionally periodic electron system

The tunneling current between an electron gas with a periodic potential in two dimensions and a plain two-dimensional electron system (2DES) has been studied. The strength of the periodic potential, the subband energy of the plain 2DES, and an applied in-plane magnetic field were varied, mapping the Fourier transform of the periodic wave function. Periodic peaks were observed and explained by translations in the reciprocal lattice. When the potential was strongly modulated to form an array of antidots, commensurability peaks were seen in lateral transport, but, as expected, not in tunneling.

Strong enhancement of drag and dissipation at the weak- to strong-coupling phase transition in a bilayer system at a total Landau level filling nu = 1

We consider a bilayer electronic system at a total Landau level filling factor nu = 1, and focus on the transition from the regime of weak interlayer coupling to that of the strongly coupled (1,1,1) phase (or "quantum Hall ferromagnet"). Making the assumption that in the transition region the system is made of puddles of the (1,1,1) phase embedded in a bulk of the weakly coupled state, we show that the transition is accompanied by a strong increase in longitudinal Coulomb drag that reaches a maximum of approximately h/2e(2). In that regime the longitudinal drag increases with decreasing temperature.

Hard correlation gap observed in quench-condensed ultrathin beryllium

We report on the tunneling density of states (DOS) in strongly disordered ultrathin Be films quench condensed at 20 K. Above 5 K, the DOS shows the well-known logarithmic anomaly at the Fermi level. Only in a narrow temperature range near 2 K is the DOS linearly dependent on energy, as predicted by Efros and Shklovskii. However, both the zero-bias conductance and the slope of the linear DOS are found to decrease drastically with decreasing temperature. Tunneling measurements at mK temperatures have revealed conclusively that a hard correlation gap opens up in the DOS.

Observation of a linearly dispersing collective mode in a quantum hall ferromagnet

Double-layer two-dimensional electron systems can exhibit a fascinating collective phase believed to display both quantum ferromagnetism and excitonic superfluidity. This unusual phase has recently been found to exhibit tunneling phenomena reminiscent of the Josephson effect. A key element of the theoretical understanding of this bizarre quantum fluid is the existence of linearly dispersing Goldstone collective modes. Using the method of tunneling spectroscopy, we have demonstrated the existence of these modes. We find the measured velocity to be in reasonable agreement with theoretical estimates.

Theory of interlayer tunneling in bilayer quantum Hall ferromagnets

Spielman et al. [Phys. Rev. Lett. 84, 5808 (2000] recently observed a large and sharp Josephson-like zero-bias peak in the tunnel conductance of a bilayer system in a quantum Hall ferromagnet state. We argue that disorder-induced topological defects in the pseudospin order parameter limit the peak size and destroy the predicted Josephson effect. We predict that the peak would be split and shifted by an in-plane magnetic field in a way that maps the dispersion relation of the ferromagnet's Goldstone mode. We also predict resonant structures in the dc I-V characteristic under bias by an ac electric field.

Resonantly enhanced tunneling in a double layer quantum hall ferromagnet

The tunneling conductance between two parallel 2D electron systems has been measured in a regime of strong interlayer Coulomb correlations. At total Landau level filling nuT=1 the tunnel spectrum changes qualitatively when the boundary separating the compressible phase from the ferromagnetic quantized Hall state is crossed. A huge resonant enhancement replaces the strongly suppressed equilibrium tunneling characteristic of weakly coupled layers. The possible relationship of this enhancement to the Goldstone mode of the broken symmetry ground state is discussed.