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We study nonequilibrium Andreev resonances in a voltage-biased graphene three-terminal Josephson junction (JJ). We observe periodic oscillations of resistance with maxima at multiples of the magnetic flux quantum (noninversion regime). As we increase the bias voltage, we further observe a transition point beyond which oscillations exhibit a $\pi$ phase shift (inversion regime) with maxima of resistance occuring at multiples of half-flux quantum. At this transition point, the frequency of the oscillations is doubled. We develop a model based on the coupling of the static Andreev bound states (ABSs) to the nonequilibrium Fermi surface of graphene to explain the observed noninversion to inversion crossovers. Our model associates these crossovers to microscopic phase-sensitive Andreev reflections which couple the normal and superfluid components of the current. Our findings show that multiterminal JJs can be used to engineer unconventional energy-phase relations such as those expected in the $\pi$-shifted ABSs without relying on quartet and Floquet physics. These nonequilibrium ABSs could potentially find applications in superconducting $\pi$ qubits.

Engineering superlattices (SLs) - which are spatially periodic potential landscapes for electrons - is an emerging approach for the realization of exotic properties, including superconductivity and correlated insulators, in two-dimensional materials. While moir\'e SL engineering has been a popular approach, nanopatterning is an attractive alternative offering control over the pattern and wavelength of the SL. However, the disorder arising in the system due to imperfect nanopatterning is seldom studied. Here, by creating a square lattice of nanoholes in the $SiO_2$ dielectric layer using nanolithography, we study the superlattice potential and the disorder formed in hBN-graphene-hBN heterostructures. Specifically, we observe that while electrical transport shows distinct superlattice satellite peaks, the disorder of the device is significantly higher than graphene devices without any SL. We use finite-element simulations combined with a resistor network model to calculate the effects of this disorder on the transport properties of graphene. We consider three types of disorder: nanohole size variations, adjacent nanohole mergers, and nanohole vacancies. Comparing our experimental results with the model, we find that the disorder primarily originates from nanohole size variations rather than nanohole mergers in square SLs. We further confirm the validity of our model by comparing the results with quantum transport simulations. Our findings highlight the applicability of our simple framework to predict and engineer disorder in patterned SLs, specifically correlating variations in the resultant SL patterns to the observed disorder. Our combined experimental and theoretical results could serve as a valuable guide for optimizing nanofabrication processes to engineer disorder in nanopatterned SLs.

A Josephson junction, formed between two phase-biased superconductors and a normal metal, hosts a discrete spectrum of Andreev bound states (ABS). In this paper, we develop a theory for long ballistic Andreev interferometers in two-dimensional metals. We consider three frameworks in our theoretical analysis: (i) perturbation theory in the tunneling amplitudes; (ii) non-perturbative transport theory; and (iii) physically motivated approximations to visualize the conductance maps in the (flux, voltage) plane. We find a non-standard phase-sensitive Andreev reflection process in ballistic interferometers that couples the supercurrent to the non-equilibrium populations of the ABS in the normal region. Furthermore, our model shows that conductance spectroscopy follows the spectrum of the ABS in long junctions. We also discuss our results in terms of the semiclassical theory, the classical orbits being the one-dimensional Andreev tubes. Our theoretical analysis captures the results of recent experiments by the Penn State and Harvard groups., Comment: 32 pages, 16 figures, second revision, further physical discussion added

Nonreciprocal superconducting devices have attracted growing interest in recent years as they potentially enable directional charge transport for applications in superconducting quantum circuits. Specifically, the superconducting diode effect has been explored in two-terminal devices that exhibit superconducting transport in one current direction while showing dissipative transport in the opposite direction. Here, we exploit multi-terminal Josephson junctions (MTJJs) to engineer magnetic-field-free nonreciprocity in multi-port networks. We show that when treated as a two-port electrical network, a three-terminal Josephson junction (JJ) with an asymmetric graphene region exhibits reconfigurable two-port nonreciprocity. We observe nonreciprocal (reciprocal) transport between superconducting terminals with broken (preserved) spatial mirror symmetry. We explain our observations by considering a circuit-network of JJs with different critical currents.

There is growing interest in using multi-terminal Josephson junctions (MTJJs) as a platform to artificially emulate topological phases and to investigate complex superconducting mechanisms such as quartet and multiplet Cooper pairings. Current experimental signatures in MTJJs have led to conflicting interpretations of the salient features. In this work, we report a collaborative experimental and theoretical investigation of graphene-based four-terminal Josephson junctions. We observe resonant features in the differential resistance maps that resemble those ascribed to multiplet Cooper pairings. To understand these features, we model our junctions using a circuit network of coupled two-terminal resistively and capacitively shunted junctions (RCSJs). Under appropriate bias current, the model predicts that a current flowing between two diagonal terminals in a four-terminal geometry may be represented as a sinusoidal function of a weighted sum of the superconducting phases. We show that starting from a semi-classical model with diffusive current-phase relations, the MTJJ effectively emulates a general form of the expected current-phase relation for multiplet Cooper pairings. Our study therefore suggests that differential resistance measurements alone are insufficient to conclusively distinguish resonant Andreev reflection processes from semi-classical circuit-network effects., Comment: 9 pages, 8 figures

We investigate the integer quantum Hall effect in Cd3As2 thin films under conditions of strong to moderate quantum confinement (thicknesses of 10 nm, 12 nm, 15 nm). In all the films, we observe the integer quantum Hall effect in the spin-polarized lowest Landau level (filling factor {\nu} = 1) and at spin-degenerate higher index Landau levels with even filling factors ({\nu} = 2,4,6). With increasing quantum confinement, we also observe a lifting of the Landau level spin degeneracy at {\nu} = 3, manifest as the emergence of an anomaly in the longitudinal and Hall resistivity. Tight-binding calculations show that the enhanced g-factor likely arises from a combination of quantum confinement and corrections from nearby subbands. We also comment on the magnetic field induced transition from an insulator to a quantum Hall liquid when the chemical potential is near the charge neutrality point.

In quantum anomalous Hall (QAH) insulators, the interior is insulating but electrons can travel with zero resistance along one-dimensional conducting paths known as chiral edge channels (CECs). These CECs have been predicted to be confined to the one-dimensional (1D) edges and exponentially decay in the two-dimensional (2D) bulk. In this work, we present the results of a systematic study of QAH devices fashioned in a Hall bar geometry of different widths. At the charge neutral point, the QAH effect persists in a Hall bar device with a width of only ~72 nm, implying the intrinsic decaying length of CECs is less than ~36 nm. In the electron-doped regime, we find that the Hall resistance deviates quickly from the quantized value when the sample width is less than 1 um. Our theoretical calculations suggest that the deviation from the quantized Hall resistance in narrow QAH samples originates from the interaction between two opposite CECs mediated by disorder-induced bulk states in QAH insulators, consistent with our experimental observations., Comment: 19 pages, 5 figures, comments are welcome

The charge and exchange statistics of an elementary excitation manifest in quantum coherent oscillations that can be explored in interferometry measurements. Quantum Hall interferometers are primary tools to uncover unconventional quantum statistics associated with fractional and non-Abelian anyons of a two-dimensional system, the latter being the foundation of topological quantum computing. Graphene interferometers offer new avenues to explore the physics of exotic excitations due to their relatively small charging energies and sharp confinement potentials. Bilayer graphene possesses a true band gap to facilitate the formation of quantum confinement and exhibits the most robust even-denominator fractional quantum Hall states that may host non-Abelian anyons. Here we present the design and fabrication of a split-gated bilayer graphene Fabry-P\'erot interferometer and experimental evidence of Aharonov-Bohm interference at multiple integer quantum Hall states. The versatility of the device allows us to study a wide range of scenarios, determine the velocities of edge states, and assess dephasing mechanisms of the interferometer. These results pave the way to the quest of non-Abelian statistics in this promising device platform., Comment: 14 pages, 3 figures, and 2 tables

A quantum anomalous Hall (QAH) insulator is a topological state of matter, in which the interior is insulating but electrical current flows along the edges of the sample, in either clockwise (right-handed) or counter-clockwise (left-handed) direction dictated by the spontaneous magnetization orientation. Such chiral edge current (CEC) eliminates any backscattering, giving rise to quantized Hall resistance and zero longitudinal resistance. In this work, we fabricate mesoscopic QAH sandwich (i.e. magnetic topological insulator (TI)/TI/magnetic TI) Hall bar devices and succeed in switching the CEC chirality in QAH insulators through spin-orbit torque (SOT) by applying a current pulse and suitably controlled gate voltage. The well-quantized QAH states with opposite CEC chiralities are demonstrated through four- and three-terminal measurements before and after SOT switching. Our theoretical calculations show that the SOT that enables the magnetization switching can be generated by both bulk and surface carriers in QAH insulators, in good agreement with experimental observations. Current pulse-induced switching of the CEC chirality in QAH insulators will not only advance our knowledge in the interplay between magnetism and topological states but also expedite easy and instantaneous manipulation of the QAH state in proof-of-concept energy-efficient electronic and spintronic devices as well as quantum information applications., Comment: 24 pages, 5 figures, comments are welcome

We use magnetotransport in dual-gated magnetic topological insulator heterostructures to map out a phase diagram of the topological Hall and quantum anomalous Hall effects as a function of the chemical potential (primarily determined by the back gate voltage) and the asymmetric potential (primarily determined by the top gate voltage). A theoretical model that includes both surface states and valence band quantum well states allows the evaluation of the variation of the Dzyaloshinskii-Moriya interaction and carrier density with gate voltages. The qualitative agreement between experiment and theory provides strong evidence for the existence of a topological Hall effect in the system studied, opening up a new route for understanding and manipulating chiral magnetic spin textures in real space., Comment: Minor corrections to original submission

A quantum anomalous Hall (QAH) insulator coupled to an s-wave superconductor is predicted to harbor a topological superconducting phase, the elementary excitations of which (i.e. Majorana fermions) can form topological qubits upon non-Abelian braiding operations. A recent transport experiment interprets the half-quantized two-terminal conductance plateau as the presence of chiral Majorana fermions in a millimeter-size QAH-Nb hybrid structure. However, there are concerns about this interpretation because non-Majorana mechanisms can also generate similar signatures, especially in a disordered QAH system. Here, we fabricated QAH-Nb hybrid structures and studied the QAH-Nb contact transparency and its effect on the corresponding two-terminal conductance. When the QAH film is tuned to the metallic regime by electric gating, we observed a sharp zero-bias enhancement in the differential conductance, up to 80% at zero magnetic field. This large enhancement suggests high probability of Andreev reflection and transparent interface between the magnetic topological insulator (TI) and Nb layers. When the magnetic TI film is in the QAH state with well-aligned magnetization, we found that the two-terminal conductance is always half-quantized. Our experiment provides a comprehensive understanding of the superconducting proximity effect observed in QAH-superconductor hybrid structures and shows that the half-quantized conductance plateau is unlikely to be induced by chiral Majorana fermions., Comment: 17 pages, 4 figures. Comments are welcome. Supplementary materials will be provided upon request

The quantum anomalous Hall (QAH) effect is a quintessential consequence of non-zero Berry curvature in momentum-space. The QAH insulator harbors dissipation-free chiral edge states in the absence of an external magnetic field. On the other hand, the topological Hall (TH) effect, a transport hallmark of the chiral spin textures, is a consequence of real-space Berry curvature. While both the QAH and TH effects have been reported separately, their coexistence, a manifestation of entangled chiral edge states and chiral spin textures, has not been reported. Here, by inserting a TI layer between two magnetic TI layers to form a sandwich heterostructure, we realized a concurrence of the TH effect and the QAH effect through electric field gating. The TH effect is probed by bulk carriers, while the QAH effect is characterized by chiral edge states. The appearance of TH effect in the QAH insulating regime is the consequence of chiral magnetic domain walls that result from the gate-induced Dzyaloshinskii-Moriya interaction and occur during the magnetization reversal process in the magnetic TI sandwich samples. The coexistence of chiral edge states and chiral spin textures potentially provides a unique platform for proof-of-concept dissipationless spin-textured spintronic applications., Comment: 22 pages, 4 figures, the accepted version for Nature Materials

Three-dimensional topological insulators (TI's) in proximity with superconductors are expected to exhibit exotic phenomena such as topological superconductivity (TSC) and Majorana bound states (MBS), which may have applications in topological quantum computation. In superconductor-TI-superconductor Josephson junctions, the supercurrent versus the phase difference between the superconductors, referred to as the current-phase relation (CPR), reveals important information including the nature of the superconducting transport. Here, we study the induced superconductivity in gate-tunable Josephson junctions (JJs) made from topological insulator BiSbTeSe2 with superconducting Nb electrodes. We observe highly skewed (non-sinusoidal) CPR in these junctions. The critical current, or the magnitude of the CPR, increases with decreasing temperature down to the lowest accessible temperature (T ~ 20 mK), revealing the existence of low-energy modes in our junctions. The gate dependence shows that close to the Dirac point the CPR becomes less skewed, indicating the transport is more diffusive, most likely due to the presence of electron/hole puddles and charge inhomogeneity. Our experiments provide strong evidence that superconductivity is induced in the highly ballistic topological surface states (TSS) in our gate-tunable TI- based JJs. Furthermore, the measured CPR is in good agreement with the prediction of a model which calculates the phase dependent eigenstate energies in our system, considering the finite width of the electrodes as well as the TSS wave functions extending over the entire circumference of the TI.

We report anomalous enhancement of the critical current at low temperatures in gate-tunable Josephson junctions made from topological insulator BiSbTeSe$_2$ nanoribbons with superconducting Nb electrodes. In contrast to conventional junctions, as a function of the decreasing temperature $T$, the increasing critical current $I_c$ exhibits a sharp upturn at a temperature $T_*$ around 20$\%$ of the junction critical temperatures for several different samples and various gate voltages. The $I_c$ vs. $T$ demonstrates a short junction behavior for $T>T_*$, but crosses over to a long junction behavior for $T

We report on the observation of gate-tunable proximity-induced superconductivity and multiple Andreev reflections (MAR) in a bulk-insulating BiSbTeSe$_2$ topological insulator nanoribbon (TINR) Josephson junction (JJ) with superconducting Nb contacts. We observe a gate-tunable critical current ($I_C$) for gate voltages ($V_g$) above the charge neutrality point ($V_{CNP}$), with $I_C$ as large as 430 nA. We also observe MAR peaks in the differential conductance ($dI/dV$) versus DC voltage ($V_{dc}$) across the junction corresponding to sub-harmonic peaks (at $V_{dc} = V_n = 2\Delta_{Nb}/en$, where $\Delta_{Nb}$ is the superconducting gap of the Nb contacts and $n$ is the sub-harmonic order). The sub-harmonic order, $n$, exhibits a $V_g$-dependence and reaches $n = 13$ for $V_g = 40$ V, indicating the high transparency of the Nb contacts to TINR. Our observations pave the way toward exploring the possibilities of using TINR in topologically protected devices that may host exotic physics such as Majorana fermions.

We have investigated the low frequency (f) flicker (also called 1/f) noise of single-layer graphene devices on h-BN (placed on SiO2/Si) along with those on SiO2/Si. We observe that the devices fabricated on h-BN have on average one order of magnitude lower noise amplitude compared with devices fabricated on SiO2/Si. We associate this noise reduction to the lower densities of impurities and trap sites in h-BN than in SiO2. Furthermore, the gate voltage dependent noise amplitude shows a broad maximum at Dirac point for devices on h-BN, in contrast to the M-shaped behavior showing a minimum at Dirac point for devices on SiO2, consistent with the reduced charge inhomogeneity (puddles) for graphene on h-BN. This study demonstrates that the use of h-BN as a substrate or dielectric can be a simple and efficient noise reduction technique valuable for electronic applications of graphene and other nanomaterials.

Over the past few years, there has been a growing interest in layered transition metal dichalcogenides (TMD) such as molybdenum disulfide (MoS2). Most studies so far have focused on the electronic and optoelectronic properties of single-layer MoS2, whose band structure features a direct bandgap, in sharp contrast to the indirect bandgap of thicker MoS2. In this paper, we present a systematic study of the thickness-dependent electrical and thermoelectric properties of few-layer MoS2. We observe that the electrical conductivity () increases as we reduce the thickness of MoS2 and peaks at about two layers, with six-time larger conductivity than our thickest sample (23-layer MoS2). Using a back-gate voltage, we modulate the Fermi energy () of the sample where an increase in the Seebeck coefficient () is observed with decreasing gate voltage () towards the subthreshold (OFF state) of the device, reaching as large as in a four-layer MoS2. While previous reports have focused on a single-layer MoS2 and measured Seebeck coefficient in the OFF state, which has vanishing electrical conductivity and thermoelectric power factor (), we show that MoS2-based devices in their ON state can have as large as in the two-layer sample. The increases with decreasing thickness then drops abruptly from double-layer to single-layer MoS2, a feature we suggest as due to a change in the energy dependence of the electron mean-free-path according to our theoretical calculation. Moreover, we show that care must be taken in thermoelectric measurements in the OFF state to avoid obtaining erroneously large Seebeck coefficients when the channel resistance is very high. Our study paves the way towards a more comprehensive examination of the thermoelectric performance of two-dimensional (2D) semiconductors., Comment: 35 pages, 20 figures