Edge channels of broken-symmetry quantum Hall states in graphene visualized by atomic force microscopy.

The quantum Hall (QH) effect, a topologically non-trivial quantum phase, expanded the concept of topological order in physics bringing into focus the intimate relation between the "bulk" topology and the edge states. The QH effect in graphene is distinguished by its four-fold degenerate zero energy Landau level (zLL), where the symmetry is broken by electron interactions on top of lattice-scale potentials. However, the broken-symmetry edge states have eluded spatial measurements. In this article, we spatially map the quantum Hall broken-symmetry edge states comprising the graphene zLL at integer filling factors of [Formula: see text] across the quantum Hall edge boundary using high-resolution atomic force microscopy (AFM) and show a gapped ground state proceeding from the bulk through to the QH edge boundary. Measurements of the chemical potential resolve the energies of the four-fold degenerate zLL as a function of magnetic field and show the interplay of the moiré superlattice potential of the graphene/boron nitride system and spin/valley symmetry-breaking effects in large magnetic fields.

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

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

Achieving µeV tunneling resolution in an in-operando scanning tunneling microscopy, atomic force microscopy, and magnetotransport system for quantum materials research.

Research in new quantum materials requires multi-mode measurements spanning length scales, correlations of atomic-scale variables with a macroscopic function, and spectroscopic energy resolution obtainable only at millikelvin temperatures, typically in a dilution refrigerator. In this article, we describe a multi-mode instrument achieving a µeV tunneling resolution with in-operando measurement capabilities of scanning tunneling microscopy, atomic force microscopy, and magnetotransport inside a dilution refrigerator operating at 10 mK. We describe the system in detail including a new scanning probe microscope module design and sample and tip transport systems, along with wiring, radio-frequency filtering, and electronics. Extensive benchmarking measurements were performed using superconductor-insulator-superconductor tunnel junctions, with Josephson tunneling as a noise metering detector. After extensive testing and optimization, we have achieved less than 8 µeV instrument resolving capability for tunneling spectroscopy, which is 5-10 times better than previous instrument reports and comparable to the quantum and thermal limits set by the operating temperature at 10 mK.

Disorder induced power-law gaps in an insulator-metal Mott transition.

A correlated material in the vicinity of an insulator-metal transition (IMT) exhibits rich phenomenology and a variety of interesting phases. A common avenue to induce IMTs in Mott insulators is doping, which inevitably leads to disorder. While disorder is well known to create electronic inhomogeneity, recent theoretical studies have indicated that it may play an unexpected and much more profound role in controlling the properties of Mott systems. Theory predicts that disorder might play a role in driving a Mott insulator across an IMT, with the emergent metallic state hosting a power-law suppression of the density of states (with exponent close to 1; V-shaped gap) centered at the Fermi energy. Such V-shaped gaps have been observed in Mott systems, but their origins are as-yet unknown. To investigate this, we use scanning tunneling microscopy and spectroscopy to study isovalent Ru substitutions in Sr3(Ir1-xRux)2O7 (0 ≤ x ≤ 0.5) which drive the system into an antiferromagnetic, metallic state. Our experiments reveal that many core features of the IMT, such as power-law density of states, pinning of the Fermi energy with increasing disorder, and persistence of antiferromagnetism, can be understood as universal features of a disordered Mott system near an IMT and suggest that V-shaped gaps may be an inevitable consequence of disorder in doped Mott insulators.

Interaction-driven quantum Hall wedding cake-like structures in graphene quantum dots.

Quantum-relativistic matter is ubiquitous in nature; however, it is notoriously difficult to probe. The ease with which external electric and magnetic fields can be introduced in graphene opens a door to creating a tabletop prototype of strongly confined relativistic matter. Here, through a detailed spectroscopic mapping, we directly visualize the interplay between spatial and magnetic confinement in a circular graphene resonator as atomic-like shell states condense into Landau levels. We directly observe the development of a "wedding cake"-like structure of concentric regions of compressible-incompressible quantum Hall states, a signature of electron interactions in the system. Solid-state experiments can, therefore, yield insights into the behavior of quantum-relativistic matter under extreme conditions.

Interplay of orbital effects and nanoscale strain in topological crystalline insulators.

Orbital degrees of freedom can have pronounced effects on the fundamental properties of electrons in solids. In addition to influencing bandwidths, gaps, correlation strength and dispersion, orbital effects have been implicated in generating novel electronic and structural phases. Here we show how the orbital nature of bands can result in non-trivial effects of strain on band structure. We use scanning-tunneling microscopy to study the effects of strain on the electronic structure of a heteroepitaxial thin film of a topological crystalline insulator, SnTe. By studying the effects of uniaxial strain on the band structure we find a surprising effect where strain applied in one direction has the most pronounced influence on the band structure along the perpendicular direction. Our theoretical calculations indicate that this effect arises from the orbital nature of the conduction and valence bands. Our results imply that a microscopic model capturing strain effects must include a consideration of the orbital nature of bands.

An on/off Berry phase switch in circular graphene resonators.

The phase of a quantum state may not return to its original value after the system's parameters cycle around a closed path; instead, the wave function may acquire a measurable phase difference called the Berry phase. Berry phases typically have been accessed through interference experiments. Here, we demonstrate an unusual Berry phase-induced spectroscopic feature: a sudden and large increase in the energy of angular-momentum states in circular graphene p-n junction resonators when a relatively small critical magnetic field is reached. This behavior results from turning on a π Berry phase associated with the topological properties of Dirac fermions in graphene. The Berry phase can be switched on and off with small magnetic field changes on the order of 10 millitesla, potentially enabling a variety of optoelectronic graphene device applications.

Helical Level Structure of Dirac Potential Wells.

In graphene and other massless two-dimensional Dirac materials, Klein tunneling compromises electron confinement, and momentum-space contours can be assigned a Berry phase which is either zero or π. Consequently, in such systems the energy spectrum of circular potential wells exhibits an interesting discontinuity as a function of magnetic field B: for a given angular momentum the ladder of eigen-resonances is split at an energy-dependent critical field B c. Here we show that introducing a mass term Δ in the Hamiltonian bridges this discontinuity in such a way that states below B c are adiabatically connected to states above B c whose principal quantum number differs by unity depending on the sign of Δ. In the B-Δ plane, the spectrum of these circular resonators resembles a spiral staircase, in which a particle prepared in the ∣n, m⟩ resonance state can be promoted to the ∣n± 1, m⟩ state by an adiabatic circuit of the Hamiltonian about B c, the sign depending on the direction of the circuit. We explain the phenomenon in terms of the evolving Berry phase of the orbit, which in such a circuit changes adiabatically by 2π.

Strain engineering Dirac surface states in heteroepitaxial topological crystalline insulator thin films.

The unique crystalline protection of the surface states in topological crystalline insulators has led to a series of predictions of strain-generated phenomena, from the appearance of pseudo-magnetic fields and helical flat bands to the tunability of Dirac surface states by strain that may be used to construct 'straintronic' nanoswitches. However, the practical realization of this exotic phenomenology via strain engineering is experimentally challenging and is yet to be achieved. Here, we have designed an experiment to not only generate and measure strain locally, but also to directly measure the resulting effects on Dirac surface states. We grew heteroepitaxial thin films of topological crystalline insulator SnTe in situ and measured them using high-resolution scanning tunnelling microscopy to determine picoscale changes in the atomic positions, which reveal regions of both tensile and compressive strain. Simultaneous Fourier-transform scanning tunnelling spectroscopy was then used to determine the effects of strain on the Dirac electrons. We find that strain continuously tunes the momentum space position of the Dirac points, consistent with theoretical predictions. Our work demonstrates the fundamental mechanism necessary for using topological crystalline insulators in strain-based applications.

Nanoscale determination of the mass enhancement factor in the lightly doped bulk insulator lead selenide.

Bismuth chalcogenides and lead telluride/selenide alloys exhibit exceptional thermoelectric properties that could be harnessed for power generation and device applications. Since phonons play a significant role in achieving these desired properties, quantifying the interaction between phonons and electrons, which is encoded in the Eliashberg function of a material, is of immense importance. However, its precise extraction has in part been limited due to the lack of local experimental probes. Here we construct a method to directly extract the Eliashberg function using Landau level spectroscopy, and demonstrate its applicability to lightly doped thermoelectric bulk insulator PbSe. In addition to its high energy resolution only limited by thermal broadening, this novel experimental method could be used to detect variations in mass enhancement factor at the nanoscale level. This opens up a new pathway for investigating the local effects of doping and strain on the mass enhancement factor.

Dirac mass generation from crystal symmetry breaking on the surfaces of topological crystalline insulators.

The tunability of topological surface states and controllable opening of the Dirac gap are of fundamental and practical interest in the field of topological materials. In the newly discovered topological crystalline insulators (TCIs), theory predicts that the Dirac node is protected by a crystalline symmetry and that the surface state electrons can acquire a mass if this symmetry is broken. Recent studies have detected signatures of a spontaneously generated Dirac gap in TCIs; however, the mechanism of mass formation remains elusive. In this work, we present scanning tunnelling microscopy (STM) measurements of the TCI Pb1-xSnxSe for a wide range of alloy compositions spanning the topological and non-topological regimes. The STM topographies reveal a symmetry-breaking distortion on the surface, which imparts mass to the otherwise massless Dirac electrons-a mechanism analogous to the long sought-after Higgs mechanism in particle physics. Interestingly, the measured Dirac gap decreases on approaching the trivial phase, whereas the magnitude of the distortion remains nearly constant. Our data and calculations reveal that the penetration depth of Dirac surface states controls the magnitude of the Dirac mass. At the limit of the critical composition, the penetration depth is predicted to go to infinity, resulting in zero mass, consistent with our measurements. Finally, we discover the existence of surface states in the non-topological regime, which have the characteristics of gapped, double-branched Dirac fermions and could be exploited in realizing superconductivity in these materials.

Observation of Dirac node formation and mass acquisition in a topological crystalline insulator.

In topological crystalline insulators (TCIs), topology and crystal symmetry intertwine to create surface states with distinct characteristics. The breaking of crystal symmetry in TCIs is predicted to impart mass to the massless Dirac fermions. Here, we report high-resolution scanning tunneling microscopy studies of a TCI, Pb(1-x)Sn(x)Se that reveal the coexistence of zero-mass Dirac fermions protected by crystal symmetry with massive Dirac fermions consistent with crystal symmetry breaking. In addition, we show two distinct regimes of the Fermi surface topology separated by a Van-Hove singularity at the Lifshitz transition point. Our work paves the way for engineering the Dirac band gap and realizing interaction-driven topological quantum phenomena in TCIs.

Imaging the evolution of metallic states in a correlated iridate.

The Ruddlesden-Popper series of iridates (Srn+1IrnO3n+1) have been the subject of much recent attention due to the anticipation of emergent phenomena arising from the cooperative action of spin-orbit-driven band splitting and Coulomb interactions. However, an ongoing debate over the role of correlations in the formation of the charge gap and a lack of understanding of the effects of doping on the low-energy electronic structure have hindered experimental progress in realizing many of the predicted states. Using scanning tunnelling spectroscopy we map out the spatially resolved density of states in Sr3Ir2O7 (Ir327). We show that its parent compound, argued to exist only as a weakly correlated band insulator, in fact possesses a substantial ~ 130 meV charge excitation gap driven by an interplay between structure, spin-orbit coupling and correlations. We find that single-atom defects are associated with a strong electronic inhomogeneity, creating an important distinction between the intrinsic and spatially averaged electronic structure. Combined with first-principles calculations, our measurements reveal how defects at specific atomic sites transfer spectral weight from higher energies to the gap energies, providing a possible route to obtaining metallic electronic states from the parent insulating states in the iridates.