In this work we introduce a low-energy Hamiltonian for single layer and bilayer black phosphorus that describes the electronic states at the vicinity of the gamma point. The model is based on a recently proposed tight-binding description for electron and hole bands close to the Fermi level. We calculate expressions for the Landau level spectrum as function of magnetic field and in the case of bilayer black phosphorus we investigate the effect of an external bias on the electronic band gap. The results showcase the highly anisotropic character of black phosphorus and in particular for bilayer BP, the presence of bias allows for a field-induced semiconductor-metal transition.
We report that two rotated layers of blue phosphorene behave as a direct band gap semiconductor. The optical spectrum shows absorption peaks in the visible region of the spectrum and in addition the energy of these peaks can be tuned with the rotational angle. These findings makes twisted bilayer blue phosphorene a strong candidate as a solar cell or photodetection device. Our results are based on ab initio calculations of several rotated blue phosphorene layers.
Using a simple LCAO model by Harrison, we have qualitatively studied the edge state of bilayer phosphorene, which is a unit structure of the layered crystal of black phosphorus. This model successfully reproduces the isolated edge state in the bulk gap in monolayer phosphorene. In bilayer phosphorene, however, it shows that edge states are almost buried in the valence band and there is no isolated midgap edge state at the zigzag edge. Since the buried edge state works as acceptor, holes are doped from the edge state into the bulk. This gives a possible explanation for p-type conduction in undo…
Phosphorene, a monolayer of black phosphorus, is promising for nanoelectronic applications not only because it is a natural p-type semiconductor but also it possesses a layer-number dependent direct bandgap (in the range of 0.3 eV~1.5 eV). On basis of the density functional theory calculations, we investigate electronic properties of the bilayer phosphorene with different stacking orders. We find that the direct bandgap of the bilayers can vary from 0.78 - 1.04 eV with three different stacking orders. In addition, a vertical electric field can further reduce the bandgap down to 0.56 eV (at the…
We have investigated the electronic structure of charged bilayer and trilayer phoshporene using first-principles, density-functional-theory calculations. We find that the effective dielectric constant for an external electric field applied perpendicular to phosphorene layers increases with the charge density and is twice as large as in an undoped system if the electron density is around $5\times10^{13}$cm$^{-2}$. It is known that if few-layer phosphorene is placed under such an electric field, the electron band gap decreases and if the strength of the electric field is further increased, the b…
We study direct and indirect excitons in Rydberg states in phosphorene monolayers, bilayer and van der Waals (vdW) heterostructure in an external magnetic field, applied perpendicular to the monolayer or heterostructure within the framework of the effective mass approximation. Binding energies of magnetoexcitons are calculated by a numerical integration of the Schrodinger equation using the Rytova-Keldysh potential for direct magnetoexcitons and both the Rytova-Keldysh and Coulomb potentials for indirect one. The latter allows to understand the role of screening in phosphorene. We report the m…
The intrinsic carrier transport dynamics in phosphorene is theoretically examined. Utilizing a density functional theory treatment, the low-field mobility and the saturation velocity are characterized for both electrons and holes in the monolayer and bilayer structures. The analysis clearly elucidates the crystal orientation dependence manifested through the anisotropic band structure and the carrier-phonon scattering rates. In the monolayer, the hole mobility in the armchair direction is estimated to be approximately five times larger than in the zigzag direction at room temperature (460 cm$^…
A new implementation of stochastic many-body perturbation theory for periodic 2D systems is presented. The method is used to compute quasiparticle excitations in twisted bilayer phosphorene. Excitation energies are studied using stochastic $G_0W_0$ and partially self-consistent $\bar ΔGW_0$ approaches. The approach is inexpensive; it is used to study twisted systems with unit cells containing $>2,700$ atoms ($>13,500$ valence electrons), which corresponds to a minimum twisting angle of $\approx 3.1^\circ$. Twisted bilayers exhibit band splitting, increased localization and formation of localiz…
Moiré superlattices of van der Waals heterostructures provide a powerful new way to engineer the electronic structures of two-dimensional (2D) materials. Many novel quantum phenomena have emerged in different moiré heterostructures, such as correlated insulators, superconductors, and Chern insulators in graphene systems and moiré excitons in transition metal dichalcogenide (TMDC) systems. Twisted phosphorene offers another attractive system to explore moiré physics because phosphorene features an anisotropic rectangular lattice, different from the isotropic hexagonal lattice in graphene and TM…
High-level first-principles computations predict blue phosphorene bilayer to be a two-dimensional metal. This structure has not been considered before and was identified by employing a block-diagram scheme that yields the complete set of five high-symmetry stacking configurations of buckled honeycomb layers, and allows their unambiguous classification. We show that all of these stacking configurations are stable or at least metastable configurations both for blue phosphorene and gray arsenene bilayers. For blue phosphorene, the most stable stacking configuration has not yet been reported, and …
We study exchange interaction in zigzag bilayer phosphorene nanoribbons (ZBLPNRs) under a perpendicular electric field. We evaluate the spatial and electric field dependency of static spin susceptibility in real space in various configurations of magnetic impurities at zero temperature. The electronic properties of ZBLPNR in the presence of a gate voltage is also obtained. In comparison to the other two-dimensional (2D) materials such as graphene and silicene, ZBLPNR has two nearly degenerated quasiflat edge modes at the Fermi level, isolated from the bulk states. The band gap modulation of ZB…
We report on the theoretical electronic spectra of twisted phosphorene bilayers exhibiting moiré patterns, as computed by means of a continuous approximation to the moiré superlattice Hamiltonian. Our model is constructed by interpolating between effective $Γ$-point conduction- and valence-band Hamiltonians for the different stacking configurations approximately realized across the moiré supercell, formulated on symmetry grounds. We predict the realization of three distinct regimes for $Γ$-point electrons and holes at different twist angle ranges: a Hubbard regime for small twist angles $θ< 2^…
We investigate the optical absorption and transmission of few-layer phosphorene in the framework of ab initio density functional simulations and many-body perturbation theory at the level of random phase approximation. In bilayer phosphorene, the optical transition of the valence band to the conduction band appears along the armchair direction at about 0.72 eV, while it is absent along the zigzag direction. This phenomenon is consistent with experimental observations. The angle-resolved optical absorption in few-layer phosphorene shows that it is transparent when illuminated by near grazing in…
Using the tight-binding approach, we investigate the electronic properties of bilayer phosphorene (BLP) quantum dots (QDs) in the presence of perpendicular electric and magnetic fields. Since BLP consists of two coupled phosphorene layers, it is of interest to examine the layer-dependent electronic properties of BLP QDs, such as the electronic distributions over the two layers and the so-produced layer-polarization features, and to see how these properties are affected by the magnetic field and the bias potential. We find that in the absence of a bias potential only edge states are layer-polar…
Two-dimensional (2D) crystals' edge structures not only influence their overall properties but also dictate their formation due to edge-mediated synthesis and etching processes. Edges must be carefully examined because they often display complex, unexpected features at the atomic scale, such as reconstruction, functionalization, and uncontrolled contamination. Here, we examine atomic-scale edge structures and uncover reconstruction behavior in bilayer phosphorene. We use in situ transmission electron microscopy (TEM) of phosphorene/graphene specimens at elevated temperatures to minimize surfac…
The magneto-transport properties of phosphorene are investigated by employing the generalized tight-binding model to calculate the energy bands. For bilayer phosphorene, a composite magnetic and electric field is shown to induce a feature-rich Landau level (LL) spectrum which includes two subgroups of low-lying LLs. The two subgroups possess distinct features in level spacings, quantum numbers, as well as field dependencies. These together lead to anomalous quantum Hall (QH) conductivities which include a well-shape, staircase and composite quantum structures with steps having varying heights …
We investigate theoretically the Landau levels (LLs) and magneto-transport properties of phosphorene under a perpendicular magnetic field within the framework of the effective \textbf{\emph{k$\cdot$p}} Hamiltonian and tight-binding (TB) model. At low field regime, we find that the LLs linearly depend both on the LL index $n$ and magnetic field $B$, which is similar with that of conventional semiconductor two-dimensional electron gas. The Landau splittings of conduction and valence band are different and the wavefunctions corresponding to the LLs are strongly anisotropic due to the different an…
Phosphorene, a single layer of black phosphorus, is a direct-band gap two-dimensional semiconductor with promising charge and spin transport properties. The electronic band structure of phosphorene is strongly affected by the structural anisotropy of the underlying crystal lattice. We describe the relevant conduction and valence bands close to the $Γ$ point by four- and six-band (with spin) $k \cdot p$ models, including the previously overlooked interband spin-orbit coupling which is essential for studying anisotropic crystals. All the $k \cdot p$ parameters are obtained by a robust fit to {\i…
We theoretically investigate the thermoelectric properties of zigzag bilayer phosphorene nanoribbons (ZBPNR). We first, draw an analogy between the extended Su-Schrieffer-Heeger (SSH) ladder and ZBPNR edge states and obtain their corresponding band structure and wave functions analytically. Then, by applying the energy filtering method, we show that the electric power and thermoelectric efficiency of the ZBPNRs can be improved remarkably in the presence of mid-gap edge states. We also argue how to engineer the edge modes to further optimize thermoelectric power and efficiency of the system by …
Schottky barrier field-effect transistors (SBFETs) based on few and mono layer phosphorene are simulated by the non-equilibrium Green's function formalism. It is shown that scaling down the gate oxide thickness results in pronounced ambipolar I-V characteristics and significant increase of the minimal leakage current. The problem of leakage is especially severe when the gate insulator is thin and the number of layer is large, but can be effectively suppressed by reducing phosphorene to mono or bilayer. Different from two-dimensional graphene and layered dichalcogenide materials, both the ON-cu…
We have analyzed the effects of the anisotropic energy bands of phosphorene on magnetoroton branches for electrons and holes in the two Landau levels close to the band edges. We have found that the fractional quantum Hall effect gap in the lowest (highest) Landau level in conduction (valance) band is slightly larger than that for conventional semiconductor systems and therefore experimentally observable. We also found that the magnetoroton mode for both electrons and holes consists of two branches with two minima due to the anisotropy. Additionally, we show that due to the anisotropy, there is…
Two-dimensional gapped semi-Dirac (GSD) materials are systems with a finite band gap that their charge carriers behave relativistically in one direction and Schrödinger-like in the other. In the present work, we show that besides the two well-known energy bands features (curvature and chirality), the band-gap parameter also play a crucial role in the index- and magnetic field-dependence of the Landau levels (LLs) in a GSD system. We take the monolayer phosphorene as a GSD representative example to explicitly provide physical insights into the role of this parameter in determining the index- an…
By taking account of the electric-field-induced charge screening, a self-consistent calculation within the framework of the tight-binding approach is employed to obtain the electronic band structure of gated multilayer phosphorene and the charge densities on the different phosphorene layers. We find charge density and screening anomalies in single-gated multilayer phosphorene and electron-hole bilayers in dual-gated multilayer phosphorene. Due to the unique puckered lattice structure, both intralayer and interlayer charge screenings are important in gated multilayer phosphorene. We find that t…
Rippling is an inherent quality of two-dimensional materials playing an important role in determining their properties. Here, we study the effect of structural corrugations on the electronic and transport properties of monolayer black phosphorus (phosphorene) in the presence of tilted magnetic field. We follow a perturbative approach to obtain analytical corrections to the spectrum of Landau levels induced by a long-wavelength corrugation potential. We show that surface corrugations have a non-negligible effect on the electronic spectrum of phosphorene in tilted magnetic field. Particularly, t…
Zigzag phosphorene nanoribbons are metallic owing to the edge states, whose energies are inside the gap and far from the bulk bands. We show that -- through electrical manipulation of edge states -- electron propagation can be restricted to one of the ribbon edges or, in case of bilayer phosphorene nanoribbons, to one of the layers. This finding implies that edge and layer can be regarded as tunable equivalents of the spin-one-half degree of freedom, i.e., the pseudospin. In both layer- and edge-pseudospin schemes, we propose and characterize a pseudospin field-effect transistor, which can gen…
The energy spectrum and eigenstates of single-layer black phosphorous nanoribbons in the presence of perpendicular magnetic field and in-plane transverse electric field are investigated by means of a tight-binding method and the effect of different types of edges is analytically examined. A description based on a new continuous model is proposed by expansion of the tight-binding model in the long-wavelength approximation. The wavefunctions corresponding to the flatband part of the spectrum are obtained analytically and are shown to approach agree well with the numerical results from the tight-…
Phosphorene, a monolayer of black phosphorus (BP), is an elemental two-dimensional material with interesting physical properties, such as high charge carrier mobility and exotic anisotropic in-plane properties. To fundamentally understand these various physical properties, it is critically important to conduct an atomic-scale structural investigation of phosphorene, particularly regarding various defects and preferred edge configurations. However, it has been challenging to investigate mono- and few-layer phosphorene because of technical difficulties arising in the preparation of a high-qualit…
Phosphorene, a two-dimensional (2D) monolayer of black phosphorus, has attracted considerable theoretical interest, although the experimental realization of monolayer, bilayer, and few-layer flakes has been a significant challenge. Here we systematically survey conditions for liquid exfoliation to achieve the first large-scale production of monolayer, bilayer, and few-layer phosphorus, with exfoliation demonstrated at the 10-gram scale. We describe a rapid approach for quantifying the thickness of 2D phosphorus and show that monolayer and few-layer flakes produced by our approach are crystalli…
We obtain the exact wave function of a monolayer phosphorene under a low- intensity time-dependent magnetic field using the dynamical invariant method. We calculate the quantum-mechanical energy expectation value and the transition probability for a constant and an oscillatory magnetic field. For the former we observe that the Landau level energy varies linearly with the quantum numbers n and m and the magnetic field intensity B_0. No transition takes place. For the latter, we observe that the energy oscillates in time, increasing linearly with the Landau level n and m and nonlinearly with the…
We study collective plasmon excitations and screening of disordered single- and bilayer black phosphorus beyond the low energy continuum approximation. The dynamical polarizability of phosphorene is computed using a tight-binding model that properly accounts for the band structure in a wide energy range. Electron-electron interaction is considered within the Random Phase Approximation. Damping of the plasmon modes due to different kinds of disorder, such as resonant scatterers and long-range disorder potentials, is analyzed. We further show that an electric field applied perpendicular to bilay…
We theoretically investigate the electronic and magneto-optical properties of rectangular, hexangular, and triangular monolayer phosphorene quantum dots (MPQDs) utilizing the tight-binding method. The electronic states, density of states, electronic density distribution, and Laudau levels as well as the optical absorption spectrum are calculated numerically. Our calculations show that: (1) edge states appear in the band gap in all kinds of MPQDs regardless of their shapes and edge configurations due to the anisotropic electron hopping in monolayer phosphorene (MLP). Electrons in any edge state…
We investigate intra-Landau-level collective excitations in a bilayer disordered two-dimensional electron system exposed to a perpendicular magnetic field. The energy spectrum is calculated within the random phase approximation by taking into account electron-impurity scattering in the self-consistent Born approximation which includes consistent vertex corrections. Signatures of these bilayer excitations in drag and collective excitation measurements are identified.
Interaction driven integer quantum Hall effects are anticipated in graphene bilayers because of the near-degeneracy of the eight Landau levels which appear near the neutral system Fermi level. We predict that an intra-Landau-level cyclotron resonance signal will appear at some odd-integer filling factors, accompanied by collective modes which are nearly gapless and have approximate $k^{3/2}$ dispersion. We speculate on the possibility of unususal localization physics associated with these modes.
Spectral and transport properties of electrons in confined phosphorene systems are investigated in a five hopping parameter tight-binding model, using analytical and numerical techniques. The main emphasis is on the properties of the topological edge states accommodated by the quasi-flat band that characterizes the phosphorene energy spectrum. We show, in the particular case of phosphorene, how the breaking of the bipartite lattice structure gives rise to the electron-hole asymmetry of the energy spectrum. The properties of the topological edge states in the zig-zag nanoribbons are analyzed un…
Heterostructures formed between two different van der Waals materials enable interactions and functionalities absent in each component. In this work we show that vicinity to an atomically thin WSe$_2$ sheet dramatically impacts the energies of the symmetry-broken low Landau levels of bilayer graphene, possibly due to screening. We present a systematic study of the magnetic field and electrical displacement field dependences of the Landau level gaps at filling factor $ν$ = 1, 2, 3, and compare to BN encapsulated pristine bilayer graphene. The exchange-dominated energy splitting between the N = …
The zero-energy Landau level of bilayer graphene is shown to be anomalously sharp (delta-function like) against bond disorder as long as the disorder is correlated over a few lattice constants.The robustness of the zero-mode anomaly can be attributed to the preserved chiral symmetry. Unexpectedly, even when we apply a finite potential difference (i.e., an electric field) between the top and the bottom layers, the valley-split $n=0$ Landau levels remain anomalously sharp although they are now shifted away from the zero energy, while the $n=1$ Landau levels exhibit the usual behavior.
We show that topology of the low-energy band structure in bilayer graphene critically depends on mechanical deformations of the crystal which may easily develop in suspended graphene flakes. We describe the Lifshitz transition that takes place in strained bilayers upon splitting the parabollic bands at intermediate energies into several Dirac cones at the energy scale of few meV. Then, we show how this affects the electron Landau level spectra and the quantum Hall effect.
If bilayer graphene is placed in a high perpendicular magnetic field, several quantum Hall plateaus are observed at low enough temperatures. Of these, the $σ_{xy}=4ne^2/h$ sequence ($n\neq0$) is explained by standard Landau quantization, while the other integer plateaus arise due to interactions. The low-energy excitations in both cases are magnetoexcitons, whose dispersion relation depends on single- and many-body effects in a complicated manner. Analyzing the magnetoexciton modes in bilayer graphene, we find that the mixing of different Landau level transitions not only renormalizes them, bu…
We investigate the charge carrier dynamics in bilayer graphene subject to monochromatic laser irradiation within the Landau level quantization regime. Even though the radiation field does not lift the energy degeneracy of the lowest Landau levels (LLs), it nevertheless has a strong effect on the photoinduced pseudospin polarization response for higher LLs ($n\ge2$). Our results show that the photoinduced bandgaps lead to a finite response of the averaged pseudospin polarization with nontrivial oscillating behavior. It is shown that the contribution from these higher LL transitions turns out to…
The effective mass and superfluidity-normal phase transition temperature of magnetoexcitons in topological insulator bilayers are theoretically investigated. The intra-Landau-level Coulomb interaction is treated perturbatively, from which the effective magnetoexciton mass is analytically discussed. The inclusion of inter-Landau-level Coulomb interaction by more exact numerical diagonalization of the Hamiltonian brings out important modifications to magnetoexciton properties, which are specially characterized by prominent reduction in the magnetoexciton effective mass and promotion in the super…
Electronic structure of graphene monolayer-bilayer junction in a magnetic field is studied within an effective-mass approximation. The energy spectrum is characterized by interface Landau levels, i.e., the locally flat bands appearing near the boundary region, resulting in a series of characteristic peaks in the local density of states. Their energies are independent of boundary types such as zigzag or armchair. In the atomic scale, the local density of states shows a Kekulé pattern due to the valley mixing in the armchair boundary, while does not in the zigzag boundary.
In a magnetic field bilayer graphene supports an octet of zero-energy Landau levels with an extra twofold degeneracy in Landau orbitals n=0 and n=1. It is shown that this orbital degeneracy is lifted due to Coulombic quantum fluctuations of the valence band (the Dirac sea); this is a quantum effect analogous to the Lamb shift in the hydrogen atom. A detailed study is made of how these zero-energy levels evolve, with filling, into a variety of pseudo-zero-mode Landau levels in the presence of possible spin and valley breaking and Coulomb interactions, and a comparison is made with experimental …
We obtain analytical expressions for the eigenstates and the Landau level spectrum of biased graphene bilayers in a magnetic field. The calculations are performed in the context of a four-band continuum model and generalize previous approximate results. Solutions are presented for the spectrum as a function of interlayer coupling, the potential difference between the layers and the magnetic field. The explicit expressions allow us to calculate the oscillator strength and the selection rules for electric dipole transitions between the Landau states. Some transitions are significantly shifted in…
The magneto-optical properties of bilayer phosphorene is investigated by the generalized tight-binding model and the gradient approximation. The vertical inter-Landau-level transitions, being sensitive to the polarization directions, are mainly determined by the spatial symmetries of sub-envelope functions on the distinct sublattices. The anisotropic excitations strongly depend on the electric and magnetic fields. A perpendicular uniform electric field could greatly diversify the selection rule, frequency, intensity, number and form of symmetric absorption peaks. Specifically, the unusual magn…
The large lattice constant of Moire superlattice formed in twisted bilayer graphene for small twist enables observing the Landau levels splitting into Hofstadter butterflies in energy spectra for moderate magnetic field. This is expected for generic system under homogeneous bias conditions but its robustness against spatial potential fluctuations is left open question. We study the energy structure of twisted bilayer system in dependence of both, the homogeneous magnetic field and the bias voltage applied exclusively in its central part. Although the translational symmetry is broken, the energ…
The electromagnetic response of bilayer graphene in a magnetic field is studied in comparison with that of monolayer graphene. Both types of graphene turn out to be qualitatively quite similar in dielectric and screening characteristics, especially those deriving from vacuum fluctuations, but the effect is generally much more sizable for bilayers. The presence of the zero-(energy-)mode Landau levels is a feature specific to graphene. In bilayers, unlike in monolayers, the effect of the zero-mode levels becomes visible and even dominant in density response as an externally-controllable band gap…
The $E = 0$ octet of bilayer graphene in the filling factor range of -4 < $ν$ < 4 is a fertile playground for many-body phenomena, yet a Landau level diagram is missing due to strong interactions and competing quantum degrees of freedom. We combine measurements and modeling to construct an empirical and quantitative spectrum. The single-particle-like diagram incorporates interaction effects effectively and provides a unified framework to understand the occupation sequence, gap energies and phase transitions observed in the octet. It serves as a new starting point for more sophisticated calcula…
Bilayer graphene in a magnetic field supports eight zero-energy Landau levels, which, as a tunable band gap develops, evolve into two nearly-degenerate quartets separated by the band gap. A close look is made into the properties of such an isolated quartet of pseudo-zero-mode levels at half filling in the presence of an in-plane electric field and the Coulomb interaction. The pseudo-zero-mode levels turn out to support, via orbital mixing, charge carriers with induced electric dipole moment, which lead to characteristic collective excitations, pseudospin waves, with some controllable features;…
Energy versus magnetic field (Hofstadter butterfly diagram) in twisted bilayer graphene is studied theoretically. If we take the usual Landau gauge, we cannot take a finite periodicity even when the magnetic flux through a supercell is a rational number. We show that the \textit{periodic} Landau gauge, which has the periodicity in one direction, makes it possible to obtain the Hofstadter butterfly diagram. Since a supercell can be large, magnetic flux through a supercell normalized by the flux quantum can be a fractional number with a small denominator, even when a magnetic field is not extrem…
In [Phys. Rev. Lett. vol. 119, p. 133901 (2017)] it was argued that two parallel graphene layers in the presence of electron drift support unstable plasmon modes. Here we show that the predicted plasmon instability is an artifact of errors upon evaluation of graphene polarizability in the presence if electron drift. Crucial role of broken Galilean invariance and spatial dispersion of conductivity for suppression of plasmon instabilities is highlighted.
The stability of the zero-energy Landau levels in bilayer graphene against the chiral symmetric disorder is examined in the presence of the trigonal warping. Based on the tight-binding lattice model with a bond disorder correlated over several lattice constants, it is shown that among the four Landau levels per spin and per valley, two Landau levels exhibit the anomalous sharpness as in the absence of the trigonal warping, while the other two are broadened, yielding split peaks in the density of states. This can be attributed to the fact that the total chirality in each valley is +2 or -2, whi…
We study the effects of a vertical electric field on the electronic band structure and transport in multilayer phosphorene and its nanoribbons. In phosphorene, at a critical value of the vertical electric field ($E_c$), the band gap closes and the band structure undergoes a massive-to-massless Dirac fermion transition along the armchair direction. This transition is observable in quantum Hall measurements, as the power-law dependence of the Landau-level energy on the magnetic field $B$ goes from $\sim (n+1/2)B$ below $E_c$, to $\sim [(n+1/2)B]^{2/3}$ at $E_c$, to $\sim [(n+1/2)B]^{1/2}$ above …
Breakdown of the quantum Hall effect (QHE) is commonly associated with an electric field approaching the inter Landau-level (LL) Zener field, ratio of the Landau gap and cyclotron radius. Eluded in semiconducting heterostructures, in spite of extensive investigation, the intrinsic Zener limit is reported here using high-mobility bilayer graphene and high-frequency current noise. We show that collective excitations arising from electron-electron interactions are essential. Beyond a noiseless ballistic QHE regime a large superpoissonian shot noise signals the breakdown via inter-LL scattering. T…
At the low Landau filling factor $(ν)$ termination of the fractional quantum Hall effect (FQHE) series, two-dimensional electron systems (2DESs) exhibit an insulating phase that is understood as a form of pinned Wigner solid. Here we use microwave spectroscopy to probe the transition to the insulator for a wide quantum well (WQW) sample that can support single-layer or bilayer states depending on its overall carrier density, $n$. We find the insulator exhibits a resonance, which is characteristic of a bilayer solid. The resonance also reveals a pair of transitions within the solid, which are n…
Electronic properties of surface areas decoupled from graphite are studied using scanning tunnelling microscopy and spectroscopy. We show that it is possible to identify decoupled graphene monolayer, Bernal bilayer, and Bernal trilayer on graphite surface according to their tunnelling spectra in high magnetic field. The decoupled monolayer and bilayer exhibit Landau quantization of massless and massive Dirac fermions, respectively. The substrate generates a sizable band gap, ~35 meV, in the Bernal bilayer, therefore, the eightfold degenerate Landau level at the charge neutrality point is split…
Bilayer graphene in a magnetic field supports eight zero-energy Landau levels, which, as a tunable band gap develops, split into two nearly-degenerate quartets separated by the band gap. A close look is made into the properties of such an isolated quartet of pseudo-zero-mode levels at half filling in the presence of an in-plane electric field and the Coulomb interaction, with focus on revealing further controllable features in bilayer graphene. The half-filled pseudo-zero-mode levels support, via orbital level mixing, charge carriers with nonzero electric moment, which would lead to field-indu…
