Massless Dirac Fermions In Graphene Research Articles

New Manifestations of the Strong Nuclear Interactions in Solids

The discovery of global macroscopic, including optical, characteristics with the addition of one neutron made crystals LiH, LiD, Diamond, Graphene an excellent model for studying the strong nuclear interaction. New manifestations of strong interaction in solids are measurements of its dependence on the distance between nucleons, finding a neutron - electron binding energy (106 meV), which is in excellent agreement with the prediction of the theory (106,7 meV). Moreover, the creation of mass in massless Dirac fermions in graphene by strong nuclear interaction has been demonstrated for the first time.

Gate-tunable quantum pathways of high harmonic generation in graphene

Under strong laser fields, electrons in solids radiate high-harmonic fields by travelling through quantum pathways in Bloch bands in the sub-laser-cycle timescales. Understanding these pathways in the momentum space through the high-harmonic radiation can enable an all-optical ultrafast probe to observe coherent lightwave-driven processes and measure electronic structures as recently demonstrated for semiconductors. However, such demonstration has been largely limited for semimetals because the absence of the bandgap hinders an experimental characterization of the exact pathways. In this study, by combining electrostatic control of chemical potentials with HHG measurement, we resolve quantum pathways of massless Dirac fermions in graphene under strong laser fields. Electrical modulation of HHG reveals quantum interference between the multi-photon interband excitation channels. As the light-matter interaction deviates beyond the perturbative regime, elliptically polarized laser fields efficiently drive massless Dirac fermions via an intricate coupling between the interband and intraband transitions, which is corroborated by our theoretical calculations. Our findings pave the way for strong-laser-field tomography of Dirac electrons in various quantum semimetals and their ultrafast electronics with a gate control.

The theory for a 2D electron diffractometer using graphene

Electrons near the Fermi level behaving as massless Dirac fermions in graphene in (1+2)-D relativistic spacetime have been confirmed by an experiment. Using this aspect, a myriad of novel and interesting devices can be sought. In this paper, we laid out the theory for using a monolayer graphene sheet as an electron diffractometer, aiming at the determination of surface properties in materials. The key ingredient is the Mott scattering of electrons by screened Coulomb scatterers in (1+2)-D spacetime. The specific array of scatterers provided by a given surface placed in contact with a graphene sheet will induce an angular distribution for the electron scattering events, which can be properly measured through the electric current flowing to external electrodes. It can provide an in situ technique for characterizing quantum dot superlattices with a resolution of a few nanometers.

Spatial and magnetic confinement of massless Dirac fermions

The massless Dirac fermions and the ease to introduce spatial and magnetic confinement in graphene provide us unprecedented opportunity to explore confined relativistic matter in this condensed-matter system. Here we report the interplay between the confinement induced by external electric fields and magnetic fields of the massless Dirac fermions in graphene. When the magnetic length lB is larger than the characteristic length of the confined electric potential lV, the spatial confinement dominates and a relatively small critical magnetic field splits the spatial-confinement-induced atomic-like shell states by switching on a pi Berry phase of the quasiparticles. When the lB becomes smaller than the lV, the transition from spatial confinement to magnetic confinement occurs and the atomic-like shell states condense into Landau levels (LLs) of the Fock-Darwin states in graphene. Our experiment demonstrates that the spatial confinement dramatically changes the energy spacing between the LLs and generates large electron-hole asymmetry of the energy spacing between the LLs. These results shed light on puzzling observations in previous experiments, which hitherto remained unaddressed.

Microscopic theory of plasmon-enabled resonant terahertz detection in bilayer graphene

The electron gas hosted in a two-dimensional solid-state matrix, such as a quantum well or a two-dimensional van der Waals heterostructure, supports the propagation of plasma waves. Nonlinear interactions between plasma waves, due to charge conservation and current convection, generate a constant density gradient which can be detected as a dc potential signal at the boundaries of the system. This phenomenon is at the heart of a plasma-wave photodetection scheme which was first introduced by Dyakonov and Shur for electronic systems with a parabolic dispersion and then extended to the massless Dirac fermions in graphene. In this work, we develop the theory of plasma-wave photodetection in bilayer graphene, which has the peculiarity that the dispersion relation depends locally and dynamically on the intensity of the plasma wave. In our analysis, we show how quantum capacitance effects, arising from the local fluctuations of the electronic dispersion, modify the intensity of the photodetection signal. An external electrical bias, e.g. induced by top and bottom gates, can be used to control the strength of the quantum capacitance corrections, and thus the photoresponse.

Anomalous caustics and Veselago focusing in 8-Pmmn borophene p–n junctions with arbitrary junction directions

Negative refraction usually demands complex structure engineering while it is very natural for massless Dirac fermions (MDFs) across the p–n junction (PNJ), this leads to Dirac electron optics. The emergent Dirac materials may exhibit hitherto unidentified phenomenon due to their nontrivial band structures in contrast to the isotropic MDFs in graphene. Here, as a specific example, we explore the negative refraction induced caustics and Veselago focusing of tilted MDFs across 8-Pmmn borophene PNJs. To this aim, we develop a technique to effectively construct the electronic Green’s function (GF) in PNJs with arbitrary junction directions. Based on analytical discussions and numerical calculations, we demonstrate the strong dependence of interference pattern on the junction direction. As the junction direction perpendicular to the tilt direction, Veselago focusing or normal caustics (similar to that in graphene) appears resting on the doping configuration of the PNJs, otherwise anomalous caustics (different from that in graphene) occurs which is manipulated by the junction direction and the doping configuration. Finally, the developed GF technique is generally promising to uncover the unique transport of emergent MDFs, and the discovered anomalous caustics makes tilted MDFs potential applications in Dirac electron optics.

Observation of phonon peaks and electron-phonon bound states in graphene

Phonons, the fundamental vibrational modes of a crystal lattice, play a crucial role in determining electronic properties of materials through electron-phonon interaction. However, it has proved difficult to directly probe the phonon modes of materials in electrical measurements. Here, we report the observation of giant quantized phonon peaks of the K and K out-of-plane phonon in graphene monolayer in magnetic fields via tunneling spectra, which are usually used to measure local electronic properties of materials. A perpendicular magnetic field quantizes massless Dirac fermions in graphene into discrete Landau levels (LLs). We demonstrate that emission or absorption of phonons of quasiparticles in the LLs of graphene generates a new sequence of discrete states: the quantized phonon modes. In our tunneling spectra, the intensity of the observed phonon peaks is about 50 times larger than that of the LLs because that the K and K out-of-plane phonon opens an inelastic tunneling channel. We also show that it is possible to switch on off the quantized phonon modes at nanoscale by controlling interactions between graphene and the supporting substrate.

Thermodynamic properties of graphene in a magnetic field and Rashba coupling

We study the thermodynamic properties of massless Dirac fermions in graphene subjected to a uniform magnetic field B together with a Rashba coupling parameter λR. The thermodynamic functions such as the Helmholtz free energy, internal energy, heat capacity and entropy are obtained in the high temperature regime using an approach based on the zeta function. These functions will be numerically examined by considering two cases related to λR smaller or greater than B. In particular, we show that the Dulong–Petit law is verified for both cases.

Electronic and optical properties of metal decorated nitrogen-doped vacancy defects in graphene

We present a first principles study of the stability, and of the electronic and optical properties of graphene with nitrogen doped vacancies. Moreover, we use the vacancies as anchoring sites for Mg, Zn, Pd al Pt atoms and vary the concentration of defects. Decoration of the defects with metal atoms produces semi-metallic systems for any studied size of the cell, with linear bands crossing at the Fermi level. The peculiar electronic properties of massless Dirac fermions in graphene are hence kept, although with anisotropic Fermi velocities. New sharp peaks appear in the optical conductivity in the visible range, thus strongly enhancing the optical response of graphene.

Spatial confinement, magnetic localization, and their interactions on massless Dirac fermions

It is of keen interest to researchers understanding different approaches to confine massless Dirac fermions in graphene, which is also a central problem in making electronic devices based on graphene. Here, we studied spatial confinement, magnetic localization and their interactions on massless Dirac fermions in an angled graphene wedge formed by two linear graphene p-n boundaries with an angle 34. Using scanning tunneling microscopy, we visualized quasibound states temporarily confined in the studied graphene wedge. Large perpendicular magnetic fields condensed the massless Dirac fermions in the graphene wedge into Landau levels (LLs). The spatial confinement of the wedge affects the Landau quantization, which enables us to experimentally measure the spatial extent of the wave functions of the LLs. The magnetic fields induce a sudden and large increase in energy of the quasibound states because of a pi Berry phase jump of the massless Dirac fermions in graphene. Such a behavior is the hallmark of the Klein tunneling in graphene. Our experiment demonstrated that the angled wedge is a unique system with the critical magnetic fields for the pi Berry phase jump depending on distance from summit of the wedge.

Magnetic Hofstadter butterfly and its topologically quantized Hall conductance

The energy spectrum of massless Dirac fermions in graphene under two dimensional periodic magnetic modulation having square lattice symmetry is calculated. We show that the translation symmetry of the problem is similar to that of the Hofstadter or TKNN problem and in the weak field limit the tight binding energy eigenvalue equation is indeed given by Harper Hofstadter hamiltonian. We show that due to its magnetic translational symmetry the Hall conductivity can be identified as a topological invariant and hence quantized. We thus extend the idea of Quantum Hall Effect to magnetically modulated two dimensional electron system. Finally we indicate possible experimental systems where this may be verified.

Author Correction: Gate-tunable third-order nonlinear optical response of massless Dirac fermions in graphene

The authors of this Article have become aware of the following recent paper that reports the electrically tunable degenerate four-wave mixing in graphene-covered SiN waveguides: “Alexander, K., Savostianova, N. A., Mikhailov, S. A., Kuyken, B. & Van Thourhout, D. Electrically tunable optical nonlinearities in graphene-covered SiN waveguides characterized by four-wave mixing. ACS Photon. 4, 3039–3044 (2017).” This reference has been added as ref. 33 in the new sentence “Very recently, degenerate four-wave mixing in graphene-covered SiN waveguides with ion-gel gating was experimentally studied33.” Accordingly, the previous sentence has been amended to read as “Few such experiments have been reported, although they have been suggested in theoretical work28–32” and all subsequent references have been renumbered. These changes have been made in the online versions.

Gate-tunable third-order nonlinear optical response of massless Dirac fermions in graphene

Materials with massless Dirac fermions can possess exceptionally strong and widely tunable optical nonlinearities. Experiments on graphene monolayer have indeed found very large third-order nonlinear responses, but the reported variation of the nonlinear optical coefficient by orders of magnitude is not yet understood. A large part of the difficulty is the lack of information on how doping or chemical potential affects the different nonlinear optical processes. Here we report the first experimental study, in corroboration with theory, on third harmonic generation (THG) and four-wave mixing (FWM) in graphene that has its chemical potential tuned by ion-gel gating. THG was seen to have enhanced by ~30 times when pristine graphene was heavily doped, while difference-frequency FWM appeared just the opposite. The latter was found to have a strong divergence toward degenerate FWM in undoped graphene, leading to a giant third-order nonlinearity. These truly amazing characteristics of graphene come from the possibility to gate-control the chemical potential, which selectively switches on and off one- and multi-photon resonant transitions that coherently contribute to the optical nonlinearity, and therefore can be utilized to develop graphene-based nonlinear optoelectronic devices.

Phonon anomaly by massive Dirac fermions of graphene

Massless Dirac fermions in graphene can acquire a mass through different kinds of sublattice-symmetry-breaking perturbations, and there is a growing need to determine this mass using a conventional method. We describe how the mass caused by a staggered sublattice potential changes the phonon self-energy and explain the mechanism in terms of the pseudospin polarization of massive Dirac fermions. We argue that the mass is detectable using Raman spectroscopy.

Calibration of the fine-structure constant of graphene by time-dependent density-functional theory

One of the amazing properties of graphene is the ultrarelativistic behavior of its loosely bound electrons, mimicking massless fermions that move with a constant velocity, inversely proportional to a fine-structure constant ${\ensuremath{\alpha}}_{g}$ of the order of unity. The effective interaction between these quasiparticles is, however, better controlled by the coupling parameter ${\ensuremath{\alpha}}_{g}^{*}={\ensuremath{\alpha}}_{g}/\ensuremath{\epsilon}$, which accounts for the dynamic screening due to the complex permittivity $\ensuremath{\epsilon}$ of the many-valence electron system. This concept was introduced in a couple of previous studies [Reed et al., Science 330, 805 (2010) and Gan et al., Phys. Rev. B 93, 195150 (2016)], where inelastic x-ray scattering measurements on crystal graphite were converted into an experimentally derived form of ${\ensuremath{\alpha}}_{g}^{*}$ for graphene, over an energy-momentum region on the $\mathrm{eV}\phantom{\rule{0.16em}{0ex}}\AA{}{}^{\ensuremath{-}1}$ scale. Here, an accurate theoretical framework is provided for ${\ensuremath{\alpha}}_{g}^{*}$, using time-dependent density-functional theory in the random-phase approximation, with a cutoff in the interaction between excited electrons in graphene, which translates to an effective interlayer interaction in graphite. The predictions of the approach are in excellent agreement with the above-mentioned measurements, suggesting a calibration method to substantially improve the experimental derivation of ${\ensuremath{\alpha}}_{g}^{*}$, which tends to a static limiting value of $\ensuremath{\sim}0.14$. Thus, the ab initio calibration procedure outlined demonstrates the accuracy of perturbation expansion treatments for the two-dimensional gas of massless Dirac fermions in graphene, in parallel with quantum electrodynamics.

Massless Dirac fermions trapping in a quasi-one-dimensional npn junction of a continuous graphene monolayer

Massless Dirac fermions in graphene provide unprecedented opportunities to realize the Klein paradox, which is one of the most exotic and striking properties of relativistic particles. In the seminal theoretical work [Katsnelson et al., Nat. Phys. 2 620 (2006)], it was predicted that the massless Dirac fermions can pass through one-dimensional (1D) potential barriers unimpededly at normal incidence. Such a result seems to preclude confinement of the massless Dirac fermions in graphene by using 1D potential barriers. Here, we demonstrate, both experimentally and theoretically, that massless Dirac fermions can be trapped in quasi-1D npn junction of a continuous graphene monolayer. Because of highly anisotropic transmission of the massless Dirac fermions at n-p junction boundaries (the so-called Klein tunneling in graphene), charge carries incident at large oblique angles will be reflected from one edge of the junction with high probability and continue to bounce from the opposite edge. Consequently, these electrons are trapped for a finite time to form quasi-bound states in the quasi-1D npn junction. The quasi-bound states seen as pronounced resonances are probed and the quantum interference patterns arising from these states are directly visualized in our scanning tunneling microscope measurements.

Second-order nonlinear optical response of graphene

Although massless Dirac fermions in graphene constitute a centrosymmetric medium for in-plane excitations, their second-order nonlinear optical response is nonzero if the effects of spatial dispersion are taken into account. Here we present a rigorous quantum-mechanical theory of the second-order nonlinear response of graphene beyond the electric dipole approximation, which includes both intraband and interband transitions. The resulting nonlinear susceptibility tensor satisfies all symmetry and permutation properties, and can be applied to all three-wave mixing processes. We obtain useful analytic expressions in the limit of a degenerate electron distribution, which reveal quite strong second-order nonlinearity at long wavelengths, Fermi-edge resonances, and unusual polarization properties.

Observation of an anisotropic Dirac cone reshaping and ferrimagnetic spin polarization in an organic conductor.

The Coulomb interaction among massless Dirac fermions in graphene is unscreened around the isotropic Dirac points, causing a logarithmic velocity renormalization and a cone reshaping. In less symmetric Dirac materials possessing anisotropic cones with tilted axes, the Coulomb interaction can provide still more exotic phenomena, which have not been experimentally unveiled yet. Here, using site-selective nuclear magnetic resonance, we find a non-uniform cone reshaping accompanied by a bandwidth reduction and an emergent ferrimagnetism in tilted Dirac cones that appear on the verge of charge ordering in an organic compound. Our theoretical analyses based on the renormalization-group approach and the Hubbard model show that these observations are the direct consequences of the long-range and short-range parts of the Coulomb interaction, respectively. The cone reshaping and the bandwidth renormalization, as well as the magnetic behaviour revealed here, can be ubiquitous and vital for many Dirac materials.

Nonlinear plasmonics in a two-dimensional plasma layer

The nonlinear electron dynamics in a two-dimensional (2D) plasma layer are investigated theoretically and numerically. In contrast to the Langmuir oscillations in a three-dimensional (3D) plasma, a well-known feature of the 2D system is the square root dependence of the frequency on the wavenumber, which leads to unique dispersive properties of 2D plasmons. It is found that for large amplitude plasmonic waves there is a nonlinear frequency upshift similar to that of periodic gravity waves (Stokes waves). The periodic wave train is subject to a modulational instability, leading to sidebands growing exponentially in time. Numerical simulations show the breakup of a 2D wave train into localized wave packets and later into wave turbulence with immersed large amplitude solitary spikes. The results are applied to systems involving massless Dirac fermions in graphene as well as to sheets of electrons on liquid helium.

Graphene Near-Degenerate Four-Wave Mixing for Phase Characterization of Broadband Pulses in Ultrafast Microscopy

We investigate near-degenerate four-wave mixing in graphene using femtosecond laser pulse shaping microscopy. Intense near-degenerate four-wave mixing signals on either side of the exciting laser spectrum are controlled by amplitude and phase shaping. Quantitative signal modeling for the input pulse parameters shows a spectrally flat phase response of the near-degenerate four-wave mixing due to the linear dispersion of the massless Dirac Fermions in graphene. Exploiting these properties we demonstrate that graphene is uniquely suited for the intrafocus phase characterization and compression of broadband laser pulses, circumventing disadvantages of common methods utilizing second or third harmonic light.