We discuss models for massless Dirac fermions being subject to a perpendicular magnetic field in spherical geometry. These models are analogues of Haldane`s spherical construction for massful charge carriers. The single particle states constructed here are easily implemented in existing numerical code for many-body problems in conventional quantum Hall systems. Moreover, the many-body states of fully filled sublevels in the subspace of lowest Landau level index are skyrmions with respect to the layer spin.
When a magnetic skyrmion is modeled as a point particle, its dynamics depends on the precise definition of the skyrmion center. The guiding-center position, defined as the first moment of the skyrmion density, exhibits Thiele's massless dynamics; position based on the first moment of magnetization component $m_z$ shows Larmor oscillations characteristic of a massive particle. We show that, even with the latter definition, the Larmor oscillations may be absent for certain types of external forces such as adiabatic spin torque. We offer an alternative mechanical model of a skyrmion featuring two…
We study a graphene Hall probe located on top of a magnetic surface as a detector of skyrmions, using as working principle the anomalous Hall effect produced by the exchange interaction of the graphene electrons with the non-coplanar magnetization of the skyrmion. We study the magnitude of the effect as a function of the exchange interaction, skyrmion size and device dimensions. Our calculations for multiterminal graphene nanodevices, working in the ballistic regime, indicate that for realistic exchange interactions a single skyrmion would give Hall voltages well within reach of the experiment…
We explore different skyrmion types in the lowest Landau level of graphene at a filling factor $ν=\pm 1$. In addition to the formation of spin and valley pseudospin skyrmions, we show that another type of spin-valley entangled skyrmions can be stabilized in graphene due to an approximate SU(4) spin-valley symmetry that is affected by sublattice symmetry-breaking terms. These skyrmions have a clear signature in spin-resolved density measurements on the lattice scale, and we discuss the expected patterns for the different skyrmion types.
Skyrmions are topologically protected spin textures, characterized by a topological winding number N , that occur spontaneously in some magnetic materials. Recent experiments have demonstrated the capability to grow graphene on top Fe/Ir, a system that exhibits a two dimensional Skyrmion lattice. Here we show that a weak exchange coupling between the Dirac electrons in graphene and a two dimensional Skyrmion lattice with $N = \pm 1$ drives graphene into a quantum anomalous Hall phase, with a band-gap in bulk, a Chern number ${\cal C}=2N$ and chiral edge states with perfect quantization of cond…
At half filling of the fourfold degenerate Landau levels |n| \geq 1 in graphene, the ground states are spin polarized quantum Hall states that support spin skyrmion excitations for |n| =1,2,3. Working in the Hartree-Fock approximation, we compute the excitation energy of an unbound spin skyrmion-antiskyrmion excitation as a function of the Zeeman coupling strength for these Landau levels. We find for both the bare and screened Coulomb interactions that the spin skyrmion-antiskyrmion excitation energy is lower than the excitation energy of an unbound spin 1/2 electron-hole pair in a finite rang…
Graphene exhibits quantum Hall ferromagnetism in which an approximate SU(4) symmetry involving spin and valley degrees of freedom is spontaneously broken. We construct a set of integer and fractional quantum Hall states that break the SU(4) spin/valley symmetry, and study their neutral and charged excitations. Several properties of these ferromagnets can be evaluated analytically in the SU(4) symmetric limit, including the full collective mode spectrum at integer fillings. By constructing explicit wave functions we show that the lowest energy skyrmion states carry charge $\pm 1$ for {\em any} …
As a consequence of the approximate spin-valley symmetry in graphene, the ground state of electrons in graphene at charge neutrality is a particular SU(4) quantum-Hall ferromagnet to minimize their exchange energy. If only the Coulomb interaction is taken into account, this ferromagnet can appeal either to the spin degree of freedom or equivalently to the valley pseudo-spin degree of freedom. This freedom in choice is then limited by subleading energy scales that explicitly break the SU(4) symmetry, the simplest of which is given by the Zeeman effect that orients the spin in the direction of t…
Skyrmion Excitations in the $ν=-1$ Quantum Hall state in Monolayer Graphene
Shandong Normal University（山东师范大学）；Institute of Mathematical Sciences, CIT Campus（马德拉斯理工学院数学研究所）；Homi Bhabha National Institute（霍米·巴巴国家研究所）；University of Kentucky（肯塔基大学）
Quantum Hall states that result from interaction induced lifting of the eight-fold degeneracy of the zeroth Landau level in bilayer graphene are considered. We show that at even filling factors electric charge is injected into the system in the form of charge $2e$ skyrmions. This is a rare example of binding of charges in a system with purely repulsive interactions. We calculate the skyrmion energy and size as a function of the Zeeman interaction, and discuss signatures of the charge $2e$ skyrmions in the scanning tunneling microscopy experiments.
A graphene bilayer in a transverse magnetic field has a set of Landau levels with energies $E=\pm \sqrt{N(N+1)}\hslash ω_{c}^{\ast}$ where $ω_{c}^{\ast}$ is the effective cyclotron frequency and $% N=0,1,2,...$ All Landau levels but N=0 are four times degenerate counting spin and valley degrees of freedom. The Landau level N=0 has an extra degeneracy due to the fact that orbitals $n=0$ and $n=1$ both have zero kinetic energies. At integer filling factors, Coulomb interactions produce a set of broken-symmetry states with partial or full alignement in space of the valley and orbital pseudospins.…
Due to Klein tunneling, electrostatic potentials are unable to confine Dirac electrons. We show that it is possible to confine massless Dirac fermions in a monolayer graphene sheet by inhomogeneous magnetic fields. This allows one to design mesoscopic structures in graphene by magnetic barriers, e.g. quantum dots or quantum point contacts.
Electron transport through short, phase-coherent metal-graphene-metal devices occurs via resonant transmission through particle-in-a-box-like states defined by the atomically-sharp metal leads. we study the spectrum of particle-in-a-box states for single- and bi-layer graphene, corresponding to massless and massive two-dimensional (2d) Fermions.The density of states D as a function of particle number n shows the expected relations D(n) ~ n1/2 for massless 2d Fermions (electrons in single-layer graphene) and D(n) ~ constant for massive 2d Fermions (electrons in bi-layer graphene). The single pa…
Motivated by recent graphene transport experiments, we have undertaken a numerical study of the conductivity of disordered two-dimensional massless Dirac fermions. Our results reveal distinct differences between the cases of short-range and Coulomb randomly distributed scatterers. We speculate that this behavior is related to the Boltzmann transport theory prediction of dirty-limit behavior for Coulomb scatterers.
Graphene is a two-dimensional carbon material with a honeycomb lattice and Dirac-type low-energy spectrum. In a strong magnetic field, where Coulomb interactions dominate against disorder broadening, quantum Hall ferromagnetic states realize at integer fillings. Extending the quantum Hall ferromagnetism to the fractional filling case of massless Dirac fermions, we study the elementally charge excitations which couple with the valley degrees of freedom (so-called valley skyrmions). With the use of the density matrix renomalization group (DMRG) method, the excitation gaps are calculated and extr…
By solving two-component spinor equation for massless Dirac Fermions, we show that graphene under a periodic external magnetic field exhibits a unique energy spectrum: At low energies, Dirac Fermions are localized inside the magnetic region with discrete Landau energy levels, while at higher energies, Dirac Fermions are mainly found in non-magnetic regions with continuous energy bands originating from wavefunctions analogous to particle-in-box states of electrons. These findings offer a new methodology for the control and tuning of massless Dirac Fermions in graphene.
We show that the electric charge of the Skyrmion in the vector order parameters that characterize the quantum anomalous spin Hall state and the layer-antiferromagnet in a graphene bilayer is four and zero, respectively. The result is based on the demonstration that a vortex configuration in two broken symmetry states in bilayer graphene with the quadratic band crossing has the number of zero modes doubled relative to the single layer. The doubling can be understood as a result of Kramers' theorem implied by the "pseudo time reversal" symmetry of the vortex Hamiltonian. Disordering the quantum …
Partially filled Landau levels host competing orders, with electron solids prevailing close to integer fillings before giving way to fractional quantum Hall liquids as the Landau level fills. Here, we report the observation of an electron solid with noncolinear spin texture in monolayer graphene, consistent with solidification of skyrmions---topological spin textures characterized by quantized electrical charge. In electrical transport, electron solids reveal themselves through a rapid metal-insulator transition in the bulk electrical conductivity as the temperature is lowered, accompanied by …
The advent of Moiré materials has galvanized interest in the nature of charge carriers in topological bands. In contrast to conventional materials where charge carriers are electron-like quasiparticles, topological bands allow for more exotic possibilities where charge is carried by nontrivial topological textures, such as skyrmions. However, the real space description of skyrmions is ill-suited to address the limit of small or `baby' skyrmions which consist of an electron and a few spin flips. Here, we study the formation of the smallest skyrmions -- spin polarons, formed as bound states of a…
Monolayer graphene under a strong perpendicular field exhibit quantum Hall ferromagnetism with spontaneously broken spin and valley symmetry. The approximate SU(4) spin/valley symmetry is broken by small lattice scale effects in the central Landau level corresponding to filling factors $ν=0,\pm 1$. Notably the ground state at $ν=0$ is believed to be a canted antiferromagnetic (AF) or a ferromagnetic (F) state depending on the component of the magnetic field parallel to the layer and the strength of small anisotropies. We study the skyrmions for the filling factors $ν=\pm 1,0$ by using exact di…
We study how low-energy charge carriers scatter off periodic and linear graphene grain boundaries oriented along the zigzag direction with a periodicity three times greater than that of pristine graphene. These defects map the two Dirac points into the same position, and thus allow for intervalley scattering to occur. Starting from graphene's first-neighbor tight-binding model we show how can we compute the boundary condition seen by graphene's massless Dirac fermions at such grain boundaries. We illustrate this procedure for the 3-periodic pentagon-only grain boundary, and then work out the l…
We consider the classical motion of a massless quasi-particle in a magnetic field and under a weak electromagnetic radiation with the frequency $ω$. Due to the non-parabolic, linear energy dispersion, the particle responds not only at the frequency $ω$ but generates a broad frequency spectrum around it. The linewidth of the cyclotron resonance turns out to be very broad even in a perfectly pure material which allows one to explain recent experimental data in graphene. It is concluded that the linear response theory does not work in graphene in finite magnetic fields.
We show that new massless Dirac fermions are generated when a slowly varying periodic potential is applied to graphene. These quasiparticles, generated near the supercell Brillouin zone boundaries with anisotropic group velocity, are different from the original massless Dirac fermions. The quasiparticle wavevector (measured from the new Dirac point), the generalized pseudospin vector, and the group velocity are not collinear. We further show that with an appropriate periodic potential of triangular symmetry, there exists an energy window over which the only available states are these quasipart…
Massless Dirac particles on the helicoid are theoretically investigated. With its possible application being helical graphene, we explore how the peculiarities of Dirac particles appear on the curved, screw-symmetric surface. The zweibein is used to derive the massless Dirac equation on the helicoid, as well as general curved surfaces. Bound states of massless Dirac electrons are shown to be absent on the helicoid, and then the scattering probabilities and the phase shifts on the surface are obtained from numerically calculated wave functions. We find the local density of states and the phase …
Graphene and topological insulators (TI) possess two-dimensional Dirac fermions with distinct physical properties. Integrating these two Dirac materials in a single device creates interesting opportunities for exploring new physics of interacting massless Dirac fermions. Here we report on a practical route to experimental fabrication of graphene-Sb2Te3 heterostructure. The graphene-TI heterostructures are prepared by using a dry transfer of chemical-vapor-deposition grown graphene film. ARPES measurements confirm the coexistence of topological surface states of Sb2Te3 and Dirac π bands of grap…
The dynamics of a vortex in a thin-film ferromagnet resembles the motion of a charged massless particle in a uniform magnetic field. Similar dynamics is expected for other magnetic textures with a nonzero skyrmion number. However, recent numerical simulations revealed that skyrmion magnetic bubbles show significant deviations from this model. We show that a skyrmion bubble possesses inertia and derive its mass from the standard theory of a thin-film ferromagnet. Besides center-of-mass motion, other low energy modes are waves on the edge of the bubble traveling with different speeds in opposite…
Spontaneous symmetry breaking plays a pivotal role in many areas of physics, engendering a variety of excitations from sound modes in solids to pions in nuclear physics. Equally important excitations are solitons, nonlinear configurations of the symmetry breaking field, which can enjoy exceptional stability as in the Skyrme model of nuclear forces. Here we argue that similar models may describe magic angle graphene, a remarkable new material . When the angle between two sheets of graphene is near the magic angle of $\sim 1^\circ$, insulating behavior is observed, which gives way to superconduc…
We present extensive numerical results for the thermodynamic density of states (i.e. quantum capacitance) of a two-dimensional massless Dirac fermion fluid in a doped graphene sheet. In particular, by employing the random phase approximation, we quantify the impact of screening exerted by a metal gate located nearby a graphene flake. Finally, we calculate the spin- and circularly-symmetric Landau parameter, which can be experimentally extracted from independent measurements on the same setup of the quantum capacitance and quasiparticle velocity.
The two-dimensional electron gas in a bilayer graphene in the Bernal stacking supports a variety of uniform broken-symmetry ground states in Landau level N=0 at integer filling factors $ν\in [-3,4].$ When an electric potential difference (or bias) is applied between the layers at filling factors $ν=-1,3$, the ground state evolves from an interlayer coherent state at small bias to a state with orbital coherence at higher bias where \textit{electric} dipoles associated with the orbital pseudospins order spontaneously in the plane of the layers. In this paper, we show that by further increasing t…
Skyrmionic Transport and First Order Phase Transitions in Twisted Bilayer Graphene Quantum Hall Ferromagnet
Indian Institute of Technology, Kharagpur（印度理工学院哈格普尔分校）；S. N. Bose National Centre for Basic Sciences（S.N.玻色基础科学国家中心）；National Institute for Materials Science（日本国立材料研究所）
We propose a defect-to-edge topological quantum quench protocol that can efficiently inject electric charge from defect-core states into a chiral edge current of an induced Chern insulator. The initial state of the system is assumed to be a Mott insulator, with electrons bound to topological defects that are pinned by disorder. We show that a "critical quench" to a Chern insulator mass of order the Mott gap shunts charge from defects to the edge, while a second stronger quench can trap it there and boost the edge velocity, creating a controllable current. We apply this idea to a skyrmion charg…
The virial theorem for a system of interacting electrons in a crystal, which is described within the framework of the tight-binding model, is derived. We show that, in particular case of interacting massless electrons in graphene and other Dirac materials, the conventional virial theorem is violated. Starting from the tight-binding model, we derive the generalized virial theorem for Dirac electron systems, which contains an additional term associated with a momentum cutoff at the bottom of the energy band. Additionally, we derive the generalized virial theorem within the Dirac model using the …
When quantum flavor Hall insulator phases of itinerant fermions are disordered by strong quantum fluctuations, the condensation of skyrmion textures of order parameter fields can lead to superconductivity. In this work, we address the mechanism of skyrmion condensation by considering the scattering between (2+1)-dimensional, Weyl fermions and hedgehog type tunneling configurations of order parameters that violate the skyrmion-number conservation law. We show the quantized, flavor Hall conductivity ($σ^f_{xy}$) controls the degeneracy of topologically protected, fermion zero-modes, localized on…
In the 2016 experiment by Crossno et al. [Science 351, 1058 (2016)], electronic contribution to the thermal conductivity of graphene was found to violate the well-known Wiedemann-Franz (WF) law for metals. At liquid nitrogen temperatures, the thermal to electrical conductivity ratio of charge-neutral samples was more than 10 times higher than predicted by the WF law, what was attributed to interactions between particles leading to collective behavior described by hydrodynamics. Here we show, by adapting the handbook derivation of the WF law to the case of massless Dirac fermions, that signific…
In the case of the strongly deformed graphene, gapless graphene may turn to gapped graphene at the critical deformation. We find that, like semi-massless fermions, electrons in the deformed graphene, at the critical point, mimic the dispersions of the massless fermions in one direction and the massive fermions in the other. Our predicted dispersion formula is the generalization of the previously predicted formula. The behavior of the particle-like semi-massless fermions tunneling through a gate barrier is contrasted with that of the (pure) massless fermions tunneling through a gate barrier in …
The fractal spectrum of magnetic minibands (Hofstadter butterfly), induced by the moiré super- lattice of graphene on an hexagonal crystal substrate, is known to exhibit gapped Dirac cones. We show that the gap can be closed by slightly misaligning the substrate, producing a hierarchy of conical singularities (Dirac points) in the band structure at rational values Phi = (p/q)(h/e) of the magnetic flux per supercell. Each Dirac point signals a switch of the topological quantum number in the connected component of the quantum Hall phase diagram. Model calculations reveal the scale invariant cond…
We investigate the quantum depinning of a weakly driven skyrmion out of an impurity potential in a mesoscopic magnetic insulator. For small barrier height, the Magnus force dynamics dominates over the inertial one, and the problem is reduced to a massless charged particle in a strong magnetic field. The universal form of the WKB exponent, the rate of tunneling, and the crossover temperature between thermal and quantum tunneling is provided, independently of the detailed form of the pinning potential. The results are discussed in terms of macroscopic parameters of the insulator Cu2OSeO3 and var…
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 (FW…
It is shown that systems described by Harper's equation exhibit a Dirac point at the center of the spectrum whenever the field parameter is a fraction of even denominator. The Dirac point is formed by the touching of two subbands, and the physics around such point is characterized by the relative field only, as if the latter were null at the reference value. Such behavior is consistent with the nesting property conjectured by Hofstadter, and its experimental verification would give support to such hypothesis as well as the Peierls-Onsager ansatz used to arrive at Harper's equation when crystal…
Gap opening in graphene is usually discussed in terms of a semiconducting-like spectrum, where the appearance of a finite gap at the Dirac point is accompanied by a finite mass for the fermions. In this Letter we propose a gap scenario from graphene which preserves the massless characters of the carriers. This approach explains recent spectroscopic measurements carried out in epitaxially-grown graphene, ranging from photoemission to optical transmission.
An analytical expression is obtained for the mass of an isolated magnetic skyrmion and its linearized equation of motion. The magnetic skyrmion is viewed as a topologically protected spin-wave soliton in the magnetic ultrathin films stabilized by the interfacial-Dzyaloshinskii-Moriya interaction. The equations of motion are derived from the Landau-Lifshitz-Gilbert equation for both the skyrmion charge and magnetization centers. They are generalized Thiele equations, including the gyro-term, dissipation term, external force, acceleration term with the tensorial mass, and time derivatives of the…
Using a novel structure, consisting of two, independently contacted graphene single layers separated by an ultra-thin dielectric, we experimentally measure the Coulomb drag of massless fermions in graphene. At temperatures higher than 50 K, the Coulomb drag follows a temperature and carrier density dependence consistent with the Fermi liquid regime. As the temperature is reduced, the Coulomb drag exhibits giant fluctuations with an increasing amplitude, thanks to the interplay between coherent transport in the graphene layer and interaction between the two layers.
Whereas thermal current noise $\langle I^2 \rangle$ in typical conductors is proportional to temperature $T$, $\langle I^2 \rangle$ in graphene exhibits a nonlinear $T$ dependence due to the massless nature of individual electrons. This unique $\langle I^2 \rangle$ arising from individually massless electrons is intimately linked to the non-zero collective mass of graphene electrons; namely, $\langle I^2 \rangle$ is set by the equipartition theorem applied to the collective mass's kinetic energy, with the nonlinear $T$-dependence arising from the $T$-dependence of the collective mass. This lin…
We theoretically investigate the frictional drag induced by the Coulomb interaction between spa- tially separated massless and massive fermions at low temperatures. As a model system, we use a double-layer structure composed of a two-dimensional electron gas (2DEG) and a n-doped graphene layer. We analyze this system numerically and also present analytical formulae for the drag re- sistivity in the limit of large and small interlayer separation. Both, the temperature and density dependence are investigated and compared to 2DEG-2DEG and graphene-graphene double-layer structures. Whereas the den…
The orbital magnetic susceptibility (OMS) of a gas of noninteracting massless Dirac fermions is zero when the Fermi energy is away from the Dirac point. Making use of diagrammatic perturbation theory, we calculate exactly the OMS of massless Dirac fermions to first order in the Coulomb interaction demonstrating that it is finite and positive. Doped graphene sheets are thus unique systems in which the OMS is completely controlled by many-body effects.
Whereas massless Dirac fermions in monolayer graphene exhibit Klein tunneling when passing through a potential barrier upon normal incidence, such a barrier totally reflects massive Dirac fermions in bilayer graphene due to difference in chirality. We show that, in the presence of magnetic barriers, such massive Dirac fermions can have transmission through even at normal incidence. The general consequence of this behaviour for multilayer graphene consisting of massless and massive modes are mentioned. We also briefly discuss the effect of a bias voltage on such magnetotransport.
Recent experimental studies on graphene on hexagonal Boron Nitride (hBN) have demonstrated that hBN is not only a passive substrate that ensures superb electronic properties of graphene's carriers, but that it actively modifies their massless Dirac fermion character through a periodic moiré potential. In this work we present a theory of the plasmon excitation spectrum of massless Dirac fermions in a moiré superlattice. We demonstrate that graphene-hBN stacks offer a rich platform for plasmonics in which control of plasmon modes can occur not only via electrostatic gating but also by adjusting …
Intricate interplay between the periodicity of the lattice structure and that of the cyclotron motion gives rise to a well-known self-similar fractal structure of the energy eigenvalue, known as the Hofstadter butterfly, for an electron moving in lattice under magnetic field. Evolving from the $n=0$ Landau level, the central band of the Hofstadter butterfly is especially interesting since it may hold a key to the mysteries of the fractional quantum Hall effect observed in graphene. While the entire Hofstadter butterfly can be in principle obtained by solving Harper's equations numerically, the…
The quasi-2D electrons in graphene behave as massless fermions obeying a Dirac-Weyl equation in the low-energy regime near the two Fermi points. The stability of spin-polarized phases (SPP) in graphene is considered. The exchange energy is evaluated from the analytic pair-distribution functions, and the correlation energies are estimated via a closely similar four-component 2D electron fluid which has been investigated previously. SPPs appear for sufficiently high doping, when the exchange energy alone is considered. However, the inclusion of correlations is found to {\it suppress} the spin-ph…
Using the Lewis-Riesenfeld method of invariants we construct explicit analytical solutions for the massless Dirac equation in 2+1 dimensions describing quasi-particles in graphene. The Hamiltonian of the system considered contains some explicit time-dependence in addition to one resulting from being minimally coupled to a time-dependent magnetic field. The eigenvalue equations for the two spinor components of the Lewis-Riesenfeld invariant are found to decouple into a pair of supersymmetric invariants in a similar fashion as the known decoupling for the time-independent Dirac Hamiltonians.
The opening of a gap in single-layer graphene is often ascribed to the breaking of the equivalence between the two carbon sublattices. We show by angle-resolved photoemission spectroscopy that Ir- and Na-modified graphene grown on the Ir(111) surface presents a very large unconventional gap that can be described in terms of a phenomenological "massless" Dirac model. We discuss the consequences and differences of this model in comparison of the standard massive gap model, and we investigate the conditions under which such anomalous gap can arise from a spontaneous symmetry breaking.
MoS$_2$, a member of transition metal dichalcogenides (TMDs), recently emerged as one of the fastest growing two-dimensional materials due to its fascinating mechanical, thermal, electronic and optical properties. Unlike graphene which possesses massless Dirac fermions with ultra-high electron mobility, monolayer MoS$_2$ is a direct band gap semiconductor. An interesting question arises: Can monolayer MoS$_2$ also possess massless Dirac fermions with ultra-high electron mobility? Here, using first-principles calculations, we show that a monolayer MoS$_2$ allotrope, which consists of repeated s…
Broken symmetries in graphene affect the massless nature of its charge carriers. We present an analysis of scattering by defects in graphene in the presence of spin-orbit interactions (SOIs). A characteristic constant ratio ($\simeq 2$) of the transport to elastic times for massless electrons signals the anisotropy of the scattering. We show that SOIs lead to a drastic decrease of this ratio, especially at low carrier concentrations, while the scattering becomes increasingly isotropic. As the strength of the SOI determines the energy (carrier concentration) where this drop is more evident, thi…
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 Dir…
Graphene, a single sheet of graphite with honeycomb lattice structure, has massless carriers with tunable density and polarity. We investigate the ground state phase diagram of two graphene sheets (embedded in a dielectric) separated by distance $d$ where the top layer has electrons and the bottom layer has holes, using mean-field theory. We find that a uniform excitonic condensate occurs over a large range of carrier densities and is weakly dependent on the relative orientation of the two sheets. We obtain the excitonic gap, quasiparticle energy and the density of states. We show that both, t…
