Periodic Potential Modulation Research Articles

Moiré-Bose-Hubbard model for interlayer excitons in twisted transition metal dichalcogenide heterostructures

In bilayers of semiconducting transition metal dichalcogenides, the twist angle between layers can be used to introduce a highly regular periodic potential modulation on a length scale that is large compared to the unit cell. In such structures, correlated states can emerge, in which excitons in the heterostructure are strongly localized to the potential minima due to exciton-exciton interactions. We explore the transition between Mott and extended exciton phases in terms of a moir\'e-Bose-Hubbard Hamiltonian. Hopping and on-site interaction parameters are obtained from a Wannier representation of the interlayer-exciton wave functions, and a non-local Rytova-Keldysh model is used to attribute for the dielectric screening of excitons in the two-dimensional material. For sufficiently small exciton concentrations and substrate screening our model predicts the emergence of Mott-insulating states, establishing twisted TMD heterostructures as possible quantum simulators for bosonic many-body systems.

Transverse instability and universal decay of spin spiral order in the Heisenberg model

We analyze the intrinsic stability of spin spiral states in the two-dimensional Heisenberg model isolated from its environment. Our analysis reveals that the SU(2) symmetric point hosts a dynamic instability that is enabled by the existence of energetically favorable transverse deformations---both in real and spin space---of the spiral order. The instability is universal in the sense that it applies to systems with any spin number, spiral wave vector, and spiral amplitude. Unlike the Landau or modulational instabilities which require impurities or periodic potential modulation of an optical lattice, quantum fluctuations alone are sufficient to trigger the transverse instability. We analytically find the most unstable mode and its growth rate, and compare our analysis with phase-space methods. By adding an easy-plane exchange coupling that reduces the Hamiltonian symmetry from SU(2) to U(1), the stability boundary is shown to continuously interpolate between the modulational instability and the transverse instability. This suggests that the transverse instability is an intrinsic mechanism that hinders long-range phase coherence even in the presence of exchange anisotropy.

Moiré-Enabled Topological Superconductivity.

The search for artificial topological superconductivity has been limited by the stringent conditions required for its emergence. As exemplified by the recent discoveries of various correlated electronic states in twisted van der Waals materials, moiré patterns can act as a powerful knob to create artificial electronic structures. Here, we demonstrate that a moiré pattern between a van der Waals superconductor and a monolayer ferromagnet creates a periodic potential modulation that enables the realization of a topological superconducting state that would not be accessible in the absence of the moiré. The magnetic moiré pattern gives rise to Yu–Shiba–Rusinov minibands and periodic modulation of the Majorana edge modes that we detect using low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS). Moiré patterns and, more broadly, periodic potential modulations are powerful tools to overcome the conventional constraints for realizing and controlling topological superconductivity.

Universal properties of boundary and interface charges in continuum models of one-dimensional insulators

We study single-channel continuum models of one-dimensional insulators induced by periodic potential modulations which are either terminated by a hard wall (the boundary model) or feature a single region of dislocations and/or impurity potentials breaking translational invariance (the interface model). We investigate the universal properties of excess charges accumulated near the boundary and the interface, respectively. We find a rigorous analytic proof for the earlier observed linear dependence of the boundary charge on the phase of the periodic potential modulation as well as extend these results to the interface model. The linear dependence on the phase shows a universal value for the slope, and is intersected by discontinuous jumps by plus or minus one electron charge at the phase points where localized states enter or leave a band of extended states. Both contributions add up such that the periodicity of the excess charge in the phase over a $2\pi$-cycle is maintained. While in the boundary model this property is usually associated with the bulk-boundary correspondence, in the interface model a correspondence of scattering state and localized state contributions to the total interface charge is unveiled on the basis of the so-called nearsightedness principle.

Twist-driven wide freedom of indirect interlayer exciton emission in MoS2/WS2 heterobilayers

Theoretical and first-principal studies involving twisted two-dimensional (2D) heterobilayer transition metal dichalcogenides (TMDs) predict interlayer exciton (ILE) emission energy amplitudes (the energy difference between the lowest and the highest ILE emission energies) in the range of 100–260 meV. This can be translated into an interfacial exciton periodic potential modulation depth of 100–260 meV. However, experimental studies on twisted TMD heterobilayers have reported only a narrow depth of ILE emission of up to ∼70 meV. Here, we report a wide degree of freedom twist-angle-driven indirect ILE emission in chemical vapor deposition (CVD)-grown MoS2/WS2 vertical heterobilayers (up to ∼10%, amplitude of 120 ± 30 meV). This is attributed to the close interlayer spacing between MoS2 and WS2 courtesy of their similar hexagonal crystal symmetry with an almost similar size, coupled with interlayer spacing tuneability at various twist angles. The wide degree of freedom of ILE emission opens exciting avenues for exploring intriguing phenomena in tuneable twistronics.

Quantum Rings Engineered by Atom Manipulation.

We created hexagonal rings on a semiconductor surface by atom manipulation in a scanning tunneling microscope (STM). Our measurements reveal the generic level structure of a quantum ring, including its single ground state and doubly degenerate excited states. The ring shape leads to a periodic potential modulation and thereby a perturbation of the level structure that can be understood in analogy to band gap formation in a one-dimensional periodic potential. The modulation can be enhanced or inverted by further adding or removing atoms with the STM tip. Our results demonstrate the possibility of designing and controlling electron dynamics in a tunable periodic potential, holding promise for the construction of two-dimensional artificial lattices on a semiconductor surface.

Commensurability Oscillations in One-Dimensional Graphene Superlattices.

We report the experimental observation of commensurability oscillations (COs) in 1D graphene superlattices. The widely tunable periodic potential modulation in hBN-encapsulated graphene is generated via the interplay of nanopatterned few-layer graphene acting as a local bottom gate and a global Si back gate. The longitudinal magnetoresistance shows pronounced COs when the sample is tuned into the unipolar transport regime. We observe up to six CO minima, providing evidence for a long mean free path despite the potential modulation. Comparison to existing theories shows that small-angle scattering is dominant in hBN/graphene/hBN heterostructures. We observe robust COs persisting to temperatures exceeding T=150 K. At high temperatures, we find deviations from the predicted T dependence, which we ascribe to electron-electron scattering.

Effect of Landau quantization on linear magnetoresistance of a periodically modulated two-dimensional electron gas

The linear response of two-dimensional electron gas in a perpendicular magnetic field in the presence of a spatially dependent classically smooth electrostatic potential is studied theoretically, by application of the Kubo formula for nonlocal conductivity tensor. In the classical transport regime, a general expression for the conductivity tensor through the correlation functions of the homogeneous electron gas is derived. The quantum transport regime, when Landau quantization is essential, is studied for the case of unidirectional periodic potential modulation. Apart from the Shubnikov-de Haas oscillations, the resistivity can demonstrate quantum oscillations with larger periods and smaller amplitudes, which survive when temperature increases. These oscillations exist when the modulation amplitude considerably exceeds the cyclotron energy so the Landau subbands, formed out of the Landau levels by the modulation potential, overlap in the energy domain. Both diagonal components of the resistivity tensor demonstrate oscillations related to modification of the density of states by the modulation. In addition, the resistivity component perpendicular to the modulation axis, which is caused by the scattering-assisted hopping transport, shows another kind of oscillations related to enhancement of the hopping probability when the guiding center of cyclotron orbit shifts by the doubled cyclotron radius. It is suggested that such high-temperature oscillations can be detected under conditions when the modulation period considerably exceeds the cyclotron radius.

Spatial self-organization of macroscopic quantum states of exciton-polaritons in acoustic lattices

Exciton-polariton systems can sustain macroscopic quantum states (MQSs) under a periodic potential modulation. In this paper, we investigate the structure of these states in acoustic square lattices by probing their wave functions in real and momentum spaces using spectral tomography. We show that the polariton MQSs, when excited by a Gaussian laser beam, self-organize in a concentric structure, consisting of a single, two-dimensional gap-soliton (GS) state surrounded by one dimensional (1D) MQSs with lower energy. The latter form at hyperbolical points of the modulated polariton dispersion. While the size of the GS tends to saturate with increasing particle density, the emission region of the surrounding 1D states increases. The existence of these MQSs in acoustic lattices is quantitatively supported by a theoretical model based on the variational solution of the Gross–Pitaevskii equation. The formation of the 1D states in a ring around the central GS is attributed to the energy gradient in this region, which reduces the overall symmetry of the lattice. The results broaden the experimental understanding of self-localized polariton states, which may prove relevant for functionalities exploiting solitonic objects.

High-mobility capacitively-induced two-dimensional electrons in a lateral superlattice potential.

In the presence of a lateral periodic potential modulation, two-dimensional electrons may exhibit interesting phenomena, such as a graphene-like energy-momentum dispersion, Bloch oscillations, or the Hofstadter butterfly band structure. To create a sufficiently strong potential modulation using conventional semiconductor heterostructures, aggressive device processing is often required, unfortunately resulting in strong disorder that masks the sought-after effects. Here, we report a novel fabrication process flow for imposing a strong lateral potential modulation onto a capacitively induced two-dimensional electron system, while preserving the host material quality. Using this process flow, the electron density in a patterned Si/SiGe heterostructure can be tuned over a wide range, from 4.4 × 1010 cm−2 to 1.8 × 1011 cm−2, with a peak mobility of 6.4 × 105 cm2/V·s. The wide density tunability and high electron mobility allow us to observe sequential emergence of commensurability oscillations as the density, the mobility, and in turn the mean free path, increase. Magnetic-field-periodic quantum oscillations associated with various closed orbits also emerge sequentially with increasing density. We show that, from the density dependence of the quantum oscillations, one can directly extract the steepness of the imposed superlattice potential. This result is then compared to a conventional lateral superlattice model potential.

Geometric Resonance of Composite Fermions Near the ν=1/2 Fractional Quantum Hall State.

We observe geometric resonance features of composite fermions on the flanks of the even-denominator ν=1/2 fractional quantum Hall state in high-mobility two-dimensional electron and hole systems confined to wide GaAs quantum wells and subjected to a weak, strain-induced, unidirectional periodic potential modulation. The features provide a measure of how close to ν=1/2 the system stays single-component and supports a composite fermion Fermi sea before transitioning into a ν=1/2 fractional quantum Hall state, presumably, the two-component Ψ331 state.

Improved modeling of two-dimensional transitions in dense phases on crystalline surfaces. Krypton-graphite system.

This paper presents a refined technique to describe two-dimensional phase transitions in dense fluids adsorbed on a crystalline surface. Prediction of parameters of 2D liquid-solid equilibrium is known to be an extremely challenging problem, which is mainly due to a small difference in thermodynamic functions of coexisting phases and lack of accuracy of numerical experiments in case of their high density. This is a serious limitation of various attempts to circumvent this problem. To improve this situation, a new methodology based on the kinetic Monte Carlo method was applied. The methodology involves analysis of equilibrium gas-liquid and gas-solid systems undergoing an external potential, which allows gradual shifting parameters of the phase coexistence. The interrelation of the chemical potential and tangential pressure for each system is then treated with the Gibbs-Duhem equation to obtain the point of intersection corresponding to the liquid/solid-solid equilibrium coexistence. The methodology is demonstrated on the krypton-graphite system below and above the 2D critical temperature. Using experimental data on the liquid-solid and the commensurate-incommensurate transitions in the krypton monolayer derived from adsorption isotherms, the Kr-graphite Lennard-Jones parameters have been corrected resulting in a higher periodic potential modulation.

Determination of Fermi contour and spin polarization ofν=32composite fermions via ballistic commensurability measurements

We report ballistic transport commensurability minima in the magnetoresistance of $\nu =3/2$ composite fermions (CFs). The CFs are formed in high-quality two-dimensional electron systems confined to wide GaAs quantum wells and subjected to an in-plane, unidirectional periodic potential modulation. We observe a slight asymmetry of the CF commensurability positions with respect to $\nu=3/2$, which we explain quantitatively by comparing three CF density models and concluding that the $\nu=3/2$ CFs are likely formed by the minority carriers in the upper energy spin state of the lowest Landau level. Our data also allow us to probe the shape and size of the CF Fermi contour. At a fixed electron density of $\simeq 1.8 \times 10^{11}$ cm$^{-2}$, as the quantum well width increases from 30 to 60 nm, the CFs show increasing spin-polarization. We attribute this to the enhancement of the Zeeman energy relative to the Coulomb energy in wider wells where the latter is softened because of the larger electron layer thickness. The application of an additional parallel magnetic field ($B_{||}$) leads to a significant distortion of the CF Fermi contour as $B_{||}$ couples to the CFs' out-of-plane orbital motion. The distortion is much more severe compared to the $\nu=1/2$ CF case at comparable $B_{||}$. Moreover, the applied $B_{||}$ further spin-polarizes the $\nu=3/2$ CFs as deduced from the positions of the commensurability minima.

What determines the Fermi wave vector of composite fermions?

Composite fermions (CFs), exotic particles formed by pairing an even number of flux quanta to each electron, provide a fascinating description of phenomena exhibited by interacting two-dimensional electrons at high magnetic fields. At and near Landau level filling ν=1/2, CFs occupy a Fermi sea and exhibit commensurability effects when subjected to a periodic potential modulation. We observe a pronounced asymmetry in the magnetic field positions of the commensurability resistance minima of CFs with respect to the field at ν=1/2. This unexpected asymmetry is consistent with the CFs' Fermi wave vector being determined by the minority carriers in the lowest Landau level, and suggests a breaking of the particle-hole symmetry for CFs near ν=1/2.

Miniband transport in a two-dimensional electron gas with a strong periodic unidirectional potential modulation

We study the Bloch oscillations of a two-dimensional electron gas with a strong periodic potential-modulation and miniband transport along the field at low temperatures, assuming a free motion in the transverse direction. The dependence of the current on the field, the electron density, and the temperature is investigated by using a relaxation-time approximation for inelastic scattering. For a fixed total scattering rate, the field dependence of the current is sensitive to the ratio of the elastic and inelastic scattering rates in contrast with the recent result of a multiband but otherwise similar model with a weak potential modulation.

Mapping the surface electrostatic potentials of Au(111) by using barrier-height measurements

The surface electrostatic potential of Au(111) has been evaluated by using site-specific barrier-height measurements in a scanning tunneling microscope. From the spatial variation of the barrier height, sharp enlargement and reduction of the surface potential are obtained at the upper and lower step edges, respectively, which are characterized as the Smoluchowski effect. We also observe the potential with oscillatory decay away from the step, known as the Friedel potential oscillation. In addition, the periodic potential modulation induced by the surface reconstruction is evaluated quantitatively, which is confirmed by the energy shift of the bulk electronic states.

Phase equilibrium in argon films stabilized by homogeneous surfaces and thermodynamics of two-stage melting transition

Freezing of gases adsorbed on open surfaces (e.g., graphite) and in narrow pores is a widespread phenomenon which is a subject of a large number of publications. Modeling of the gas/liquid-solid transition is usually accomplished with a molecular simulation technique. However, quantitative analysis of the gas/liquid-solid coexistence and thermodynamic properties of the solid layer still encounters serious difficulties. This is mainly due to the effect of simulation box size on the lattice constant. Since the lattice constant is a function of loading and temperature, once the ordering transition has occurred, the simulation box size must be corrected in the course of simulation according to the Gibbs-Duhem equation. A significant problem is also associated with accurate prediction of the two-dimensional liquid-solid coexistence because of a small difference in densities of coexisting phases. The aim of this study is thermodynamic analysis of the two-dimensional phase coexistence in systems involving crystal-like free of defects layers in narrow slit pores. A special attention was paid to the determination of triple point temperatures. It is shown that intrinsic properties of argon monolayer adsorbed on the graphite surface are similar to those of isolated monolayer accommodated in the slit pore having width of two argon collision diameters. Analysis of the latter system is shown to be clearer and less time-consuming than the former one, which has allowed for explanation of the experimentally observed two-stage melting transition of argon monolayer on graphite without invoking the periodic surface potential modulation and orientational transition.

Anisotropic Fermi contour of (001) GaAs electrons in parallel magnetic fields

We demonstrate a severe Fermi contour anisotropy induced by the application of a parallel magnetic field to high-mobility electrons confined to a 30-nm-wide (001) GaAs quantum well. We study commensurability oscillations, namely geometrical resonances of the electron orbits with a unidirectional, surface-strain-induced, periodic potential modulation, to directly probe the size of the Fermi contours along and perpendicular to the parallel field. Their areas are obtained from the Shubnikov-de Haas oscillations. Our experimental data agree semi-quantitatively with the results of parameter-free calculations of the Fermi contours but there are significant discrepancies.

Molecular Self-Assembly on Graphene on SiO2 and h-BN Substrates

One of the suggested ways of controlling the electronic properties of graphene is to establish a periodic potential modulation on it, which could be achieved by self-assembly of ordered molecular lattices. We have studied the self-assembly of cobalt phthalocyanines (CoPc) on chemical vapor deposition (CVD) grown graphene transferred onto silicon dioxide (SiO2) and hexagonal boron nitride (h-BN) substrates. Our scanning tunneling microscopy (STM) experiments show that, on both substrates, CoPc forms a square lattice. However, on SiO2, the domain size is limited by the corrugation of graphene, whereas on h-BN, single domain extends over entire terraces of the underlying h-BN. Additionally, scanning tunneling spectroscopy (STS) measurements suggest that CoPc molecules are doped by the substrate and that the level of doping varies from molecule to molecule. This variation is larger on graphene on SiO2 than on h-BN. These results suggest that graphene on h-BN is an ideal substrate for the study of molecular self-assembly toward controlling the electronic properties of graphene by engineered potential landscapes.

Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices

Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum. Known as Hofstadter's butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields, and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasibly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely. Here we demonstrate that moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.