The lifting of the degeneracy of the states from the graphene $n$=0 Landau level (LL) is investigated through a non-interacting tight-binding model with random hoppings. A disorder-driven splitting of two bands and of two critical energies is observed by means of density of states and participation ratio calculations. The analysis of the probability densities of the states within the $n$=0 LL provides some insights into the interplay of lattice and disorder effects on the splitting process. An uneven spatial distribution of the wave function amplitudes between the two graphene sublattices is found for the states in between the two split peaks. It is shown that as the splitting is increased (linear increasing with disorder and square root increasing with magnetic field), the two split levels also get increasingly broadened, in such a way that the proportion of the overlapped states keeps approximately constant for a wide range of disorder or magnetic field variation.
Broadening of the Landau levels in graphene and the associated quantum Hall plateau-to-plateau transition are investigated numerically. For correlated bond disorder, the graphene-specific n=0 Landau level of the Dirac fermions becomes anomalously sharp accompanied by the Hall transition exhibiting a fixed-point-like criticality. Similarly anomalous behavior for the n=0 Landau level is also shown to occur in correlated random magnetic fields, which suggests that the anomaly is generic to disorders that preserve the chiral symmetry.
We report on magneto-transport measurements on low-density, large-area monolayer epitaxial graphene devices grown on SiC. We show that the zero-energy Landau level (LL) in monolayer graphene, which is predicted to be magnetic field ($B$)-independent, can float up above the Fermi energy at low $B$. This is supported by the temperature ($T$)-driven flow diagram approximated by the semi-circle law as well as the $T$-independent point in the Hall conductivity $σ_{xy}$ near $e^2/h$. Our experimental data are in sharp contrast to conventional understanding of the zeroth LL and metallic-like behavior…
In this proceedings paper we report on a calculation of graphene's Landau levels in a magnetic field. Our calculations are based on a self-consistent Hartree-Fock approximation for graphene's massless-Dirac continuum model. We find that because of graphene's chiral band structure interactions not only shift Landau-level energies, as in a non-relativistic electron gas, but also alter Landau level wavefunctions. We comment on the subtle continuum model regularization procedure necessary to correctly maintain the lattice-model's particle hole symmetry properties.
Massless Dirac electrons in graphene fill Landau levels with energies scaled as square roots of their numbers. Coulomb interaction between electrons leads to mixing of different Landau levels. The relative strength of this interaction depends only on dielectric susceptibility of surrounding medium and can be large in suspended graphene. We consider influence of Landau level mixing on the properties of magnetoexcitons and magnetoplasmons - elementary electron-hole excitations in graphene in quantizing magnetic field. We show that, at small enough background dielectric screening, the mixing lead…
Layer stacking and crystal lattice symmetry play important roles in the band structure and the Landau levels of multilayer graphene. ABA-stacked trilayer graphene possesses mirror-symmetry-protected monolayer-like and bilayer-like band structures. Broken mirror symmetry by a perpendicular electric field therefore induces hybridization between these bands and various quantum Hall phases emerge. We experimentally explore the evolution of Landau levels in ABA-stacked trilayer graphene under electric field. We observe a variety of valley and orbital dependent Landau level evolutions. These evoluti…
In a magnetic field graphene trilayers support a characteristic multiplet of 12 zero(-energy)-mode Landau levels with a threefold degeneracy in Landau orbitals. It was earlier noted for bilayer graphene that Coulombic vacuum fluctuations, specific to graphene, lift the orbital degeneracy of such zero-energy modes and that these Lamb-shfted" orbital modes, with filling, get mixed via the Coulomb interaction. It is pointed out that analogous orbital Lamb shift and mixing of zero-mode levels can also take place, with an enriched symmetry content, in ABC-stacked trilayer graphene; and its conseque…
The electronic spectrum of three coupled graphene layers (graphene trilayers) is investigated in the presence of an external magnetic field. We obtain analytical expressions for the Landau level spectrum for both the ABA and ABC - type of stacking, which exhibit very different dependence on the magnetic field. The effect of layer asymmetry and of external gate voltages can strongly influence the properties of the system.
We study the relaxation dynamics of the electron system in graphene flakes under Landau quantization regime using a novel approach of time-resolved Raman scattering. The non-resonant character of the experiment allows us to analyze the field dependence of the relaxation rate. Our results clearly evidence sharp increase in the relaxation rate upon the resonance between the energy of the Landau transition and the G-band and shed new light on relaxation mechanism of the Landau-quantized electrons in graphene beyond the previously studied Auger scattering.
Magnetotransport measurements are performed in ultraclean (lithographically patterned) graphene nanoribbons down to 70 nm. At high magnetic fields, a fragmentation of the electronic spectrum into a Landau levels pattern with unusual features is unveiled. The singular Landau spectrum reveals large magneto-oscillations of the Fermi energy and valley degeneracy lifting. Quantum simulations suggest some disorder threshold at the origin of mixing between opposite chiral magnetic edge states and disappearance of quantum Hall effect.
We study the discrete energy spectrum of curved graphene sheets in the presence of a magnetic field. The shifting of the Landau levels is determined for complex and realistic geometries of curved graphene sheets. The energy levels follow a similar square root dependence on the energy quantum number as for rippled and flat graphene sheets. The Landau levels are shifted towards lower energies proportionally to the average deformation and the effect is larger compared to a simple uni-axially rippled geometry. Furthermore, the resistivity of wrinkled graphene sheets is calculated for different ave…
We study nonlinear optical response of Landau quantized graphene to an intense electromagnetic wave. In particular, we consider high harmonic generation process. It is shown that one can achieve efficient generation of high harmonics with strong radiation fields -- when the work of the wave electric field on the magnetic length is larger than pump photon energy. At that high harmonics generation process takes place for a wide range of the pump wave frequencies and intensities even for significant broadening of Landau levels because of impurities in graphene.
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 observed broken-symmetry quantum Hall effects and level crossings between spin- and valley- resolved Landau levels (LLs) in Bernal stacked trilayer graphene. When the magnetic field was tilted with respect to sample normal from $0^{\circ}$ to $66^\circ$, the LL crossings formed at intersections of zeroth and second LLs from monolayer-graphene-like and bilayer-graphene-like subbands, respectively, exhibited a sequence of transitions. The results indicate the LLs from different subbands are coupled by in-plane magnetic fields ($B_{\parallel}$), which was explained by developing the tight-bind…
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.
We demonstrate the use of a quantum transport model to study heavily graded graphene p-n junctions in the quantum Hall regime. A combination of p-n interface roughness and delta function disorder potential allows us to compare experimental results on different devices from the literature. We find that wide p-n junctions suppress mixing of $n \neq 0$ Landau levels. Our simulations spatially resolve carrier transport in the device, for the first time, revealing separation of higher order Landau levels in strongly graded junctions, which suppresses mixing.
We report on polarization selection rules of inter-Landau level transitions using reflection-type optical Hall effect measurements from 600 to 4000 cm-1 on epitaxial graphene grown by thermal decomposition of silicon carbide. We observe symmetric and anti-symmetric signatures in our data due to polarization preserving and polarization mixing inter-Landau level transitions, respectively. From field-dependent measurements we identify that transitions in decoupled graphene mono-layers are governed by polarization mixing selection rules, whereas transitions in coupled graphene mono-layers are gove…
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 = …
In inhomogeneously strained graphene, low-energy electrons experience a valley-antisymmetric pseudo-magnetic field which leads to the formation of localized states at the edge between the valence and conduction bands, understood in terms of peculiar n=0 pseudo-magnetic Landau levels. Here we show that such states can manifest themselves as an isolated quadruplet of low-energy conductance resonances in a suspended stretched graphene ribbon, where clamping by the metallic contacts results in a strong inhomogeneity of strain near the ribbon ends.
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.
We show that valley degeneracy in rotationally faulted multilayer graphene may be broken in the presence of a magnetic field and interlayer commensurations. This happens due to a simultaneous breaking of both time-reversal and inversion symmetries leading to a splitting of Landau levels linear in the field. Our theoretical work is motivated by an experiment [Y. J. Song et al., Nature 467, 185 (2010)] on epitaxially grown multilayer graphene where such linear splitting of Landau levels was observed at moderate fields. We consider both bilayer and trilayer configurations and, although a linear s…
We propose Landau levels as a probe for the topological character of electronic bands in two-dimensional moiré superlattices. We consider two configurations of twisted double bilayer graphene (TDBG) that have very similar band structures, but show different valley Chern numbers of the flat bands. These differences between the AB-AB and AB-BA configurations of TDBG clearly manifest as different Landau level sequences in the Hofstadter butterfly spectra calculated using the tight-binding model. The Landau level sequences are explained from the point of view of the distribution of orbital magneti…
It is quite easy to control spin polarization and spin direction of a system via magnetic fields. However, there is no such a direct and efficient way to manipulate valley pseudospin degree of freedom. Here, we demonstrate experimentally that it is possible to realize valley polarization and valley inversion in graphene by using both strain-induced pseudomagnetic fields and real magnetic fields. The pseudomagnetic fields, which are quite different from real magnetic fields, pointing in opposite directions at the two distinct valleys of graphene. Therefore, coexistence of the pseudomagnetic fie…
In the presence of strong magnetic fields the electronic bandstructure of graphene drastically changes. The Dirac cone collapses into discrete non-equidistant Landau levels, which can be externally tuned by changing the magnetic field. In contrast to conventional materials, specific Landau levels are selectively addressable using circularly polarized light. Exploiting these unique properties, we propose the design of a tunable laser operating in the technologically promising terahertz spectral range. To uncover the many-particle physics behind the emission of light, we perform a fully quantum …
We present the results of an experimental study of the interaction of quantized Landau level (LL) edge-states at the physical edge of graphene by using a graphene pn junction device with a ring-shaped geometry for the channel. The unique device geometry allows the interactions between edge-states to be probed at both electrostatic edges defined by pn junctions and at the graphene physical edge. Measurements show that while the lowest LL edge-state is decoupled from the other LLs along the electrostatic junction, all the edge-states strongly equilibrate at the graphene physical edge despite the…
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…
Here we address the concerns of Svintsov and Ryzhii [arXiv:1812.03764] on our article on negative Landau damping in graphene [Phys. Rev. Lett. 119, 133901 (2017)]. We prove that due to the differences between the kinetic and canonical momenta, the conductivity of drift-current biased graphene is ruled by a Galilean transformation when the electron-electron interactions predominate and force the electron gas to move with constant velocity, similar to a moving medium. Furthermore, it is shown that the nonlocal effects in graphene neither preclude a negative Landau damping nor the emergence of in…
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…
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.
We report results of exact diagonalization studies of the spin- and valley-polarized fractional quantum Hall effect in the $N=0$ and 1 Landau levels in graphene. We use an effective model that incorporates Landau level mixing to lowest-order in the parameter $κ= \frac{e^2/ε\ell}{\hbar v_F/\ell}=\frac{e^2}{εv_F\hbar}$ which is magnetic field independent and can only be varied through the choice of substrate. We find Landau level mixing effects are negligible in the $N=0$ Landau level for $κ\lesssim 2$. In fact, the lowest Landau level projected Coulomb Hamiltonian is a better approximation to t…
We predict the general feasibility and demonstrate the specific design of the THz laser operating between Landau levels in graphene placed on a polar substrate in a magnetic field of order 1 T. Steady state operation under a continuous wave optical pumping is possible due to an interplay between Auger and surface-phonon mediated relaxation of carriers. The scheme is scalable to other materials with massless Dirac fermions, for example surface states in 3D topological insulators such as Bi$_2$Se$_3$ or Bi$_2$Te$_3$.
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 …
As a canonical response to the applied magnetic field, the electronic states of a metal are fundamentally reorganized into Landau levels. In Dirac metals, Landau levels can be expected without magnetic fields, provided that an inhomogeneous strain is applied to spatially modulate electron hoppings in a similar way as the Aharonov-Bohm phase. We here predict that a twisted zigzag nanoribbon of graphene exhibits strain-induced pseudo Landau levels of unexplored but analytically solvable dispersions at low energies. The presence of such dispersive pseudo Landau levels results in a negative strain…
We consider the interplay between the antiferromagnetic and Kekulé valence bond solid orderings in the zero energy Landau levels of neutral monolayer and bilayer graphene. We establish the presence of Wess-Zumino-Witten terms between these orders: this implies that their quantum fluctuations are described by the deconfined critical theories of quantum spin systems. We present implications for experiments, including the possible presence of excitonic superfluidity in bilayer graphene.
Electronic correlations stemming from nearly flat bands in van der Waals materials have demonstrated to be a powerful playground to engineer artificial quantum matter, including superconductors, correlated insulators and topological matter. This phenomenology has been experimentally observed in a variety of twisted van der Waals materials, such as graphene and dichalcogenide multilayers. Here we show that spontaneously buckled graphene can yield a correlated state, emerging from an elastic pseudo Landau level. Our results build on top of recent experimental findings reporting that, when placed…
When electrons are confined in a two dimensional (2D) system, typical quantum mechanical phenomena such as Landau quantization can be detected. Graphene systems, including the single atomic layer and few-layer stacked crystals, are ideal 2D materials for studying a variety of quantum mechanical problems. In this article, we review the experimental progress in the unusual Landau quantized behaviors of Dirac fermions in monolayer and multilayer graphene by using scanning tunneling microscopy(STM) and scanning tunneling spectroscopy(STS). Through STS measurement of the strong magnetic fields, dis…
We consider the problem of multilayer graphene on a Haldane sphere and determine the Landau level spectrum for this family of systems. This serves as a generalization of the Landau quantization problem of ordinary non-relativistic Haldane sphere and spherical graphene, or Dirac-like particles on a sphere. The Hamiltonian is diagonalized in a concise algebraic fashion exploiting two mutually commuting SU(2) algebras of the problem. Additionally, using exact wave functions we demonstrate computation of Haldane pseudopotentials in the second Landau level. These exact solutions add to the current …
We develop a rigorous quantum-mechanical theory of the nonlinear optical process of difference frequency generation of surface plasmon-polaritons in Landau-quantized graphene. Although forbidden in the electric-dipole approximation, the second-order susceptibility is surprisingly high, equivalent to the bulk magnitude above $10^{-3}$ m/V. We consider the graphene monolayer as a nonlinear optical component of a monolithic photonic chip with integrated pump fields. The nonlinear power conversion efficiency of the order of tens $μ$W/W$^2$ is predicted from structures of $10-100$ $μ$m size. We inv…
In strong magnetic fields, massless electrons in graphene populate relativistic Landau levels with the square-root dependence of each level energy on its number and magnetic field. Interaction-induced deviations from this single-particle picture were observed in recent experiments on cyclotron resonance and magneto-Raman scattering. Previous attempts to calculate such deviations theoretically using the unscreened Coulomb interaction resulted in overestimated many-body effects. This work presents many-body calculations of cyclotron and magneto-Raman transitions in single-layer graphene in the p…
We study the Landau levels in curved graphene sheets by measuring the discrete energy spectrum in the presence of a magnetic field. We observe that in rippled graphene sheets, the Landau energy levels satisfy the same square root dependence on the energy quantum number as in flat sheets, $E_n \sim \sqrt{n}$. Though, we find that the Landau levels in curved sheets are shifted towards lower energies by an amount proportional to the average spatial deformation of the sheet. Our findings are relevant for the quantum Hall effect in curved graphene sheets, which is directly related to Landau quantiz…
We show that, when graphene is subjected to an appropriate one-dimensional external periodic potential, additional branches of massless fermions are generated with nearly the same electron-hole crossing energy as that at the original Dirac point of graphene. Because of these new zero-energy branches, the Landau levels at charge neutral filling becomes 4(2N+1)-fold degenerate (with N=0,1,2,..., tunable by the potential strength and periodicity) with the corresponding Hall conductivity $σ_{xy}$ showing a step of size 4(2N+1)$e^2/h$. These theoretical findings are robust against variations in the…
In strained graphene, lattice deformation can create pseudo-magnetic fields and result in zero-field Landau level-like quantization. In the presence of an external magnetic field, valley-polarized Landau levels are predicted to be observed because the pseudo-magnetic fields are of opposite directions in the K and K' valleys of graphene. Here, we present experimental spectroscopic measurements by scanning tunneling microscopy of strained graphene on Rh foil. We direct observed valley splitting of the Landau level induced by the coexistence of the pseudo-magnetic fields and external magnetic fie…
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…
The energy-dependence of the electronic scattering time is probed by Landau level spectroscopy in quasi neutral multilayer epitaxial graphene. From the Landau levels broadening we find that the scattering rate increases linearly with energy. This implies a surprising property of the Landau level spectrum in graphene - the number of the resolved Landau levels remains constant with the applied magnetic field. Insights are given about possible scattering mechanism and carrier mobilities in the graphene system investigated.
In a magnetic field graphene trilayers support a special multiplet of 12 zero(-energy)-mode Landau levels with a threefold degeneracy in Landau orbitals. A close look is made into such zero-mode levels in ABA-stacked trilayers, with the Coulomb interaction taken into account. It turns out that the zero-mode Landau levels of ABA trilayers are greatly afflicted with electron-hole and valley asymmetries, which come from general hopping parameters and which are enhanced by the Coulomb interaction and the associated vacuum effect, the orbital Lamb shift, that lifts the zero-mode degeneracy. These a…
