We investigate the band structure of twisted monolayer-bilayer graphene (tMBG), or twisted graphene on bilayer graphene (tGBG), as a function of twist angles and perpendicular electric fields in search of optimum conditions for achieving isolated nearly flat bands. Narrow bandwidths comparable or smaller than the effective Coulomb energies satisfying $U_{\textrm{eff}} /W \gtrsim 1$ are expected for twist angles in the range of $0.3^{\circ} \sim 1.5^{\circ}$, more specifically in islands around $θ\sim 0.5^{\circ}, \, 0.85^{\circ}, \,1.3^{\circ}$ for appropriate perpendicular electric field magnitudes and directions. The valley Chern numbers of the electron-hole asymmetric bands depend intrinsically on the details of the hopping terms in the bilayer graphene, and extrinsically on factors like electric fields or average staggered potentials in the graphene layer aligned with the contacting hexagonal boron nitride substrate. This tunability of the band isolation, bandwidth, and valley Chern numbers makes of tMBG a more versatile system than twisted bilayer graphene for finding nearly flat bands prone to strong correlations.
In this brief report, we show the equivalence between the tight-binding descriptions of the monolayer and bilayer honeycomb lattices. With appropriate value of the third nearest neighbors coupling, the Hamiltonian for a monolayer is equivalent to the low energy effective Hamiltonian for bilayer in the presence of trigonal warping. A simple physical argument is provided to explain this correspondance.
Electron's tunneling through potential barrier in monolayer and bilayer graphene lattices is investigated by using full tight-binding model. Emphasis is placed on the resonance tunneling feature and inter-valley scattering probability. It is shown that normal incidence transmission probabilities for monolayer and bilayer graphene exhibit different properties. Our calculation indicates that valleytronics in graphene systems may be detected, generated and controlled by changing the structure parameters of the external electric potential.
Closed expressions for the Green functions of the stationary two-dimensional two-component Schrodinger equation for an electron moving in monolayer and bilayer graphene in the presence of a magnetic field are obtained in terms of the Whittaker functions.
This work investigates the electronic properties of the energy spectrum of a hybrid system composed of (i) a circular quantum dot of monolayer graphene surrounded by an infinite sheet of AA-stacked bilayer graphene and (ii) a circular quantum dot of AA-stacked graphene bilayer surrounded by infinite monolayer graphene. We establish analytical findings for the related energy levels and wave functions using the continuum model and the zigzag boundary conditions at the graphene monolayer-bilayer interface. We investigate the effects of perpendicular magnetic, dot radius, and electrical fields on …
The electron transmission between monolayer and bilayer graphene is theoretically studied for zigzag and armchair boundaries within an effective-mass scheme. Due to the presence of an evanescent wave in the bilayer graphene, traveling modes are well connected to each other. The transmission through the boundary is strongly dependent on the incident angle and the dependence is opposite between the K and K' points, leading to valley polarization of transmitted wave.
On the basis of tight-binding lattice model, the edge states of monolayer and bilayer graphene nanoribbons (GNRs) with different edge terminations are studied. The effects of edge-hopping modulation, spin-orbital coupling (SOC), and bias voltage on bilayer GNRs are discussed. We observe the following: (i) Some new extra edge states can be created by edge-hopping modulation for monolayer GNRs. (ii) Intralayer Rashba SOC plays a role in depressing the band energy gap $E_g$ opened by intrinsic SOC for both monolayer and bilayer GNRs. An almost linear dependent relation, i.e., $E_g\sim λ_R$, is fo…
An improved composite-boson theory of quantum Hall ferromagnets is formulated both for the monolayer and bilayer systems. In this scheme the field operator describes solely the physical degrees of freedom representing the deviation from the ground state. Skyrmions are charged excitations confined to the lowest Landau level. By evaluating the excitation energy of one skyrmion in the interlayer-coherent phase it is shown that the bilayer QH state becomes stabler as the interlayer density difference becomes larger.
Electrons behave like a classical fluid with a momentum distribution function that varies slowly in space and time when the quantum mechanical carrier-carrier scattering dominates over all other scattering processes. Recent experiments in monolayer and bilayer graphene have reported signatures of such hydrodynamic electron behavior in ultra-clean devices. In this theoretical work, starting from a microscopic treatment of electron-electron, electron-phonon and electron-impurity interactions within the Random Phase Approximation, we demonstrate that monolayer and bilayer graphene both host two d…
The tight-binding model of electrons in graphene is reviewed. We derive low-energy Hamiltonians supporting massless Dirac-like chiral fermions and massive chiral fermions in monolayer and bilayer graphene, respectively, and we describe how their chirality is manifest in the sequencing of plateaus observed in the integer quantum Hall effect. The opening of a tuneable band gap in bilayer graphene in response to a transverse electric field is described, and we explain how Hartree theory may be used to develop a simple analytical model of screening.
We use the T-matrix approximation to analyze the effect of a localized impurity on the local density of states in mono- and bilayer graphene. For monolayer graphene the Friedel oscillations generated by intranodal scattering obey an inverse-square law, while the internodal ones obey an inverse law. In the Fourier transform this translates into a filled circle of high intensity in the center of the Brillouin zone, and empty circular contours around its corners. For bilayer graphene both types of oscillations obey an inverse law.
We present an overview of the theoretical understanding of Hofstadter butterflies in monolayer and bilayer graphene. After a brief introduction on the past work in conventional semiconductor systems, we discuss the novel electronic properties of monolayer and bilayer graphene that helped to detect experimentally the fractal nature of the energy spectrum. We have discussed the theoretical background on the Moiré pattern in graphene. This pattern was crucial in determining the butterfly structure. We have also touched upon the role of electron-electron interaction in the butterfly pattern in gra…
We study the electronic spin relaxation effect in the hole-doped monolayer and bilayer transition-metal dichalcogenides in the presence of the crystal defects. We consider realistic models of the lattice vacancy and actually estimate the spin relaxation rate using the multi-orbital tight-binding model. In the monolayer, the spin-relaxation time is found to be extremely long compared to the momentum relaxation time, and this is attributed to the fact that the spin hybridization in the band structure is suppressed by the mirror reflection symmetry. The bilayer TMD has a much shorter spin relaxat…
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.
The morphology and electronic structure of pentacene (Pn) deposited on Cu(111) was studied using scanning tunneling microscopy (STM) and spectroscopy (STS). Deposition of a multilayer followed by annealing to reduce coverage to a monolayer results in the formation of either of two unique phases: a 2D herringbone structure previously unobserved for any linear acene, or a 'random- tiling' structure. Coverage greater than a monolayer promotes the formation of a bilayer phase similar to that observed for Pn/Ag(111). STS shows that the electronic structure of the first layer is strongly modified du…
We calculate the Fermi energy dependence of the (time-averaged) current and shot noise in an impurity-free carbon bilayer (length $L\ll$ width $W$), and compare with known results for a monolayer. At the Dirac point of charge neutrality, the bilayer transmits as two independent monolayers in parallel: Both current and noise are resonant at twice the monolayer value, so that their ratio (the Fano factor) has the same 1/3 value as in a monolayer -- and the same value as in a diffusive metal. The range of Fermi energies around the Dirac point within which this pseudo-diffusive result holds is sma…
Cyclotron resonance in graphene is studied with focus on many-body corrections to the resonance energies, which evade Kohn's theorem. The genuine many-body corrections turn out to derive from vacuum polarization, specific to graphene, which diverges at short wavelengths. Special emphasis is placed on the need for renormalization, which allows one to determine many-body corrections uniquely from one resonance to another. For bilayer graphene, in particular, both intralayer and interlayer coupling strengths undergo infinite renormalization; as a result, the renormalized velocity and interlayer c…
We prepare twist-controlled resonant tunneling transistors consisting of monolayer (Gr) and Bernal bilayer (BGr) graphene electrodes separated by a thin layer of hexagonal boron nitride (hBN). The resonant conditions are achieved by closely aligning the crystallographic orientation of the graphene electrodes, which leads to momentum conservation for tunneling electrons at certain bias voltages. Under such conditions, negative differential conductance (NDC) can be achieved. Application of in-plane magnetic field leads to electrons acquiring additional momentum during the tunneling process, whic…
Graphene is a gapless semiconductor in which conduction and valence band wavefunctions differ only in the phase difference between their projections onto the two sublattices of the material's two-dimensional honeycomb crystal structure. We explain why this circumstance creates openings for broken symmetry states, including antiferromagnetic states in monolayer and bilayer graphene and exciton condensates in double-layer graphene, that are momentum space analogs of the real-space order common in systems with strong local interactions. We discuss some similarities among, and some differences bet…
We model optical absorption of monolayer and bilayer graphene on hexagonal boron nitride for the case of closely-aligned crystal lattices. We show that perturbations with different spatial symmetry can lead to similar absorption spectra. We suggest that a study of the absorption spectra as a function of the doping for almost completely full first miniband is necessary to extract meaningful information about the moire characteristics from optical absorption measurements and to distinguish between various theoretical proposals for the physically realistic interaction. Also, for bilayer graphene,…
We investigate the dispersion relation and damping of plasmon modes in a bilayer-monolayer graphene heterostructure with carrier densities and at zero temperature within the random-phase-approximation taking into account the nonhomogeneity of the dielectric background of the system. We derive analytical expressions for plasmon frequencies by using long wavelength expansion of response and bare Coulomb interaction functions. We show that optical plasmon dispersion curve of the bilayer-monolayer system lies slightly below that of double-layer graphene (DLG) and the acoustic one lies much lower t…
Perfluoropentacene (PFP), an $n$-type organic semiconductor, is deposited at monolayer and bilayer coverage on Cu(111). Scanning tunneling microscopy at various bias voltages is used to investigate the geometric and electronic structure of the layer. The appearances of the first layer and second layer differ, probably because of perturbation to the first layer electronic structure by the substrate. This has been previously observed for pentacene (Pn), the isostructural $p$-type organic semiconductor. The PFP film has a unit cell of (4 -3,3 4) relative to the substrate, which is larger than tha…
The relativistic-like behavior of electrons in graphene significantly influences the interaction properties of these electrons in a quantizing magnetic field, resulting in more stable fractional quantum Hall effect states as compared to those in conventional (non-relativistic) semiconductor systems. In bilayer graphene the interaction strength can be controlled by a bias voltage and by the orientation of the magnetic field. The finite bias voltage between the graphene monolayers can in fact, enhance the interaction strength in a given Landau level. As a function of the bias voltage, a graphene…
We show that Friedel charge oscillation near an interface opens a gap at the Fermi energy for electrons with wave vectors perpendicular to the interface. If the Friedel gaps on two sides of the interface are different, a nonequlibrium effect - shifting of these gaps under bias - leads to asymmetric transport upon reversing the bias polarity. The predicted transport asymmetry is revealed by scanning tunneling potentiometry at monolayer-bilayer interfaces in epitaxial graphene on SiC (0001). This intriguing interfacial transport behavior opens a new avenue towards novel quantum functions such as…
We investigated the valley depolarization and carrier relaxation process of monolayer and bilayer WS2 at 10 K by using time-resolved Kerr rotation (TRKR) and differential reflectance measurement simultaneously. Two decay processes extracted from TRKR signals were observed on both monolayer and bilayer WS2. In monolayer WS2, the initial ultrafast decay component (less than 1 ps) was interpreted as the stimulated emission or Pauli-blocking of electrons and holes in one valley by comparing with carrier decay process. The relatively slow component was around 4 ps under low excitation energy (less …
We study both monolayer and bilayer graphene transport properties taking into account the presence of correlations in the spatial distribution of charged impurities. In particular we find that the experimentally observed sublinear scaling of the graphene conductivity can be naturally explained as arising from impurity correlation effects in the Coulomb disorder, with no need to assume the presence of short-range scattering centers in addition to charged impurities. We find that also in bilayer graphene correlations among impurities induce a crossover of the scaling of the conductivity at highe…
We investigate the current-voltage characteristics of a field-effect tunnelling transistor comprised of both monolayer and bilayer graphene with well-aligned crystallographic axes, separated by three layers of hexagonal boron nitride. Using a self-consistent description of the device's electrostatic configuration we relate the current to three distinct tunable voltages across the system and hence produce a two-dimensional map of the I-V characteristics in the low energy regime. We show that the use of gates either side of the heterostructure offers a fine degree of control over the device's ri…
The optical conductivity of graphene and bilayer graphene in quantizing magnetic fields is studied. Both dynamical conductivities, longitudinal and Hall's, are analytically evaluated. The conductivity peaks are explained in terms of electron transitions. Correspondences between the transition frequencies and the magneto-optical features are established using the theoretical results. The main optical transitions obey the selection rule with $Δn=1$ for the Landau number $n$. The Faraday rotation and light transmission in the quantizing magnetic fields are calculated. The effects of temperatures …
We report the influence of uniaxial tensile mechanical strain in the range 0-2.2% on the phonon spectra and bandstructures of monolayer and bilayer molybdenum disulfide (MoS2) two-dimensional crystals. First, we employ Raman spectroscopy to observe phonon softening with increased strain, breaking the degeneracy in the E' Raman mode of MoS2, and extract a Grüneisen parameter of ~1.06. Second, using photoluminescence spectroscopy we measure a decrease in the optical band gap of MoS2 that is roughly linear with strain, ~45 meV% strain for monolayer MoS2 and ~120 meV% strain for bilayer MoS2. Thir…
The structural and vibrational properties of metal-organic interfaces have been examined by means of infrared (IR) absorption spectroscopy and density functional theory (DFT) with an approach accounting for long-range dispersive interactions. We focus on a comparative study of the PTCDA monolayer and bilayer on Ag(111). The equilibrium geometry at the molecule-metal interface and the infrared spectrum of the chemisorbed monolayer of PTCDA on Ag(111) are well described by the computations. In the bilayer structure, the presence of a physisorbed adlayer on top of PTCDA/Ag(111) presents a challen…
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…
Proximity orbital and spin-orbit effects of bilayer graphene on monolayer WSe$_2$ are investigated from first-principles. We find that the built-in electric field induces an orbital band gap of about 10 meV in bilayer graphene. Remarkably, the proximity spin-orbit splitting for holes is two orders of magnitude---the spin-orbit splitting of the valence band at K is about 2 meV---more than for electrons. Effectively, holes experience spin-valley locking due to the strong proximity of the lower graphene layer to WSe$_2$. However, applying an external transverse electric field of some 1 V/nm, coun…
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$^…
We explore the device potential of tunable-gap bilayer graphene FET exploiting the possibility of opening a bandgap in bilayer graphene by applying a vertical electric field via independent gate operation. We evaluate device behavior using atomistic simulations based on the self-consistent solution of the Poisson and Schroedinger equations within the NEGF formalism. We show that the concept works, but bandgap opening is not strong enough to suppress band-to-band tunneling in order to obtain a sufficiently large Ion/Ioff ratio for CMOS device operation.
The strong excitonic effect in monolayer transition metal dichalcogenide (TMD) semiconductors has enabled many fascinating light-matter interaction phenomena. Examples include strongly coupled exciton-polaritons and nearly perfect atomic monolayer mirrors. The strong light-matter interaction also opens the door for dynamical control of mechanical motion through the exciton resonance of monolayer TMDs. Here we report the observation of exciton-optomechanical coupling in a suspended monolayer MoSe2 mechanical resonator. By moderate optical pumping near the MoSe2 exciton resonance, we have observ…
Results of a study on the electrochemical properties of exfoliated single and multilayer graphene flakes are presented. Graphene flakes were deposited on silicon/silicon oxide wafers to enable fast and accurate characterization by optical microscopy and Raman spectroscopy. Conductive silver paint and silver wires were used to fabricate contacts; epoxy resin was employed as masking coating in order to expose a stable, well defined area of graphene. Both multilayer and monolayer graphene microelectrodes showed quasi-reversible behavior during voltammetric measurements in potassium ferricyanide. …
The line shape of the double-resonant $2D$ Raman mode in bilayer graphene is often considered to be characteristic for a certain laser excitation energy. Here, in a joint experimental and theoretical study, we analyze the dependence of the double-resonant Raman scattering processes in bilayer graphene on the electronic broadening parameter $γ$. We demonstrate that the ratio between symmetric and anti-symmetric scattering processes sensitively depends on the lifetime of the electronic states, explaining the experimentally observed variation of the complex $2D$-mode line shape.
In many atomically thin materials their optical absorption is dominated by excitonic transitions. It was recently found that optical selection rules in these materials are influenced by the band topology near the valleys. We propose that gate-controlled band ordering in a single atomic monolayer, through changes in the valley winding number and excitonic transitions, can be probed in helicity-resolved absorption and photoluminescence. This predicted tunable band topology is confirmed by combining an effective Hamiltonian and a Bethe-Salpeter equation for an accurate description of excitons, wi…
We investigate the transport properties of double-gated bilayer graphene nanoribbons at room temperature. The devices were fabricated using conventional CMOS-compatible processes. By analyzing the dependence of the resistance at the charge neutrality point as a function of the electric field applied perpendicular to the graphene surface, we show that a band gap in the density of states opens, reaching an effective value of ~sim50 meV. This demonstrates the potential of bilayer graphene as FET channel material in a conventional CMOS environment.
We report an experimental study of temperature-dependent spectral lineshapes of phonon side-band emission stemming from dark excitons in monolayer and bilayer WSe$_{2}$. Using photoluminescence spectroscopy in the range from 4 to 100K, we observe a pronounced asymmetry in the phonon-assisted luminescence from momentum-indirect exciton reservoirs. We demonstrate that the corresponding spectral profiles are distinct from those of bright excitons with direct radiative decay pathways. The lineshape asymmetry reflects thermal distribution of exciton states with finite center-of-mass momenta, charac…
We investigate the electronic band structure and the proximity exchange effect in bilayer graphene on a family of ferromagnetic multilayers Cr$_2$X$_2$Te$_6$, X=Ge, Si, and Sn, with first principles methods. In each case the intrinsic electric field of the heterostructure induces an orbital gap on the order of 10 meV in the graphene bilayer. The proximity exchange is strongly band dependent. For example, in the case of Cr$_2$Ge$_2$Te$_6$, the low-energy valence band of bilayer graphene has exchange splitting of 8 meV, while the low energy conduction band's splitting is 30 times less (0.3 meV).…
We show that a coherent picture of the dc conductivity of monolayer and bilayer graphene at finite electronic densities emerges upon considering that strong short-range potentials are the main source of scattering in these two systems. The origin of the strong short-range potentials may lie in adsorbed hydrocarbons at the surface of graphene. The equivalence among results based on the partial-wave description of scattering, the Lippmann-Schwinger equation, and the T-matrix approach is established. Scattering due to resonant impurities close to the neutrality point is investigated via a numeric…
We have investigated the transmission in monolayer graphene barrier at nonzero angle of incidence. Taking the influence of parallel wave vector into account, the transmission as the function of incidence energy has a gap due to the evanescent waves in two cases of Klein tunneling and classical motion. The modulation of the transmission gap by the incidence angle, the height, and width of potential barrier may lead to potential applications in graphene-based electronic devices.
Electric transport of double gated bilayer graphene devices is studied as a function of charge density and bandgap. A top gate electrode can be used to control locally the Fermi level to create a pn junction between the double-gated and single-gated region. These bilayer graphene pn diodes are characterized by non-linear currents and directional current rectification, and we show the rectified direction of the source-drain voltage can be controlled by using gate voltages. A systematic study of the pn junction characteristics allows to extract a gate-dependent bandgap value which ranges from 0 …
We study the thermoelectric properties of monolayers of MoS$_{2}$ and other group-VI dichalcogenides under circularly polarized off-resonant light. Analytical expressions are derived for the Berry phase mediated magnetic moment, orbital magnetization, as well as thermal and Nernst conductivities. Tuning of the band gap by {\it off-resonant} light enhances the spin splitting in both the valence and conduction bands and, thus, leads to a dramatic improvement of the spin and valley thermoelectric properties.
We study the conductance of a biased bilayer graphene flake with monolayer nanoribbon contacts. We find that the transmission through the bilayer ribbon strongly depends on the applied bias between the two layers and on the relative position of the monolayer contacts. Besides the opening of an energy gap on the bilayer, the bias allows to tune the electronic density on the bilayer flake, making possible the control of the electronic transmission by an external parameter.
Despite the now vast body of two-dimensional materials under study, bilayer graphene remains unique in two ways: it hosts a simultaneously tunable band gap and electron density; and stems from simple fabrication methods. These two advantages underscore why bilayer graphene is critical as a material for optoelectronic applications. In the work that follows, we calculate the one- and two-photon absorption coefficients for degenerate interband absorption in a graphene bilayer hosting an asymmetry gap and adjustable chemical potential--all at finite temperature. Our analysis is comprehensive, char…
We study the collective electronic excitations in metallic single- and bilayer transition metal dichalcogenides (TMDCs) using time dependent density functional theory in the random phase approximation. For very small momentum transfers (below $q\approx0.02$~Å$^{-1}$) the plasmon dispersion follows the $\sqrt{q}$ behavior expected for free electrons in two dimensions. For larger momentum transfer the plasmon energy is significantly red shifted due to screening by interband transitions. At around $q\approx 0.1$ Å$^{-1}$ the plasmon enters the dissipative electron-hole continuum and the plasmon d…
We theoretically calculate the interaction-induced frictional Coulomb drag resistivity between two graphene monolayers as well as between two graphene bilayers, which are spatially separated by a distance "$d$". We show that the drag resistivity between graphene monolayers can be significantly affected by the intralayer momentum-relaxation mechanism. For energy independent intralayer scattering, the frictional drag induced by inter-layer electron-electron interaction goes asymptotically as $ρ_D \sim T^2/n^4d^6$ and $ρ_D \sim T^2/n^2d^2$ in the high-density ($k_F d \gg 1$) and low-density ($k_F…
Monolayer transition metal dichalcogenides, known for exhibiting strong excitonic resonances, constitute a very interesting and versatile platform for investigation of light-matter interactions. In this work we report on a strong coupling regime between excitons in monolayer WSe2 and photons confined in an open, voltage-tunable dielectric microcavity. The tunability of our system allows us to extend the exciton-polariton state over a wide energy range and, in particular, to bring the excitonic component of the lower polariton mode into resonance with other excitonic transitions in monolayer WS…
In addition to electron charge and spin, novel materials host another degree of freedom, the valley. For a junction composed of valley filter sandwiched by two normal terminals, we focus on the valley efficiency under disorder with two valley filter models based on monolayer and bilayer graphene. Applying the transfer matrix method, valley resolved transmission coefficients are obtained. We find that: i) under weak disorder, when the line defect length is over about $15\rm nm$, it functions as a perfect channel (quantized conductance) and valley filter (totally polarized); ii) in the diffusive…
We demonstrate theoretically that quantum dots in bilayers of graphene can be realized. A position-dependent doping breaks the equivalence between the upper and lower layer and lifts the degeneracy of the positive and negative momentum states of the dot. Numerical results show the simultaneous presence of electron and hole confined states for certain doping profiles and a remarkable angular momentum dependence of the quantum dot spectrum which is in sharp contrast with that for conventional semiconductor quantum dots. We predict that the optical spectrum will consist of a series of non-equidis…
Using full-potential density functional calculations within local density approximation (LDA), we found mechanically tunable band-gap in ZnO monolayer (ML-ZnO) in graphene-like honeycomb structure, by simulated application of in-plane homogeneous biaxial strain. Unstrained ML-ZnO was found to have a direct band gap of energy 1.68 eV within LDA; the actual band gap would be more, since LDA is known to underestimate the gap. Within our simulated strain limit of about plus or minus 10%, the band gap remains direct and shows a strong non-linear variation with strain. The results may find applicati…
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…
