The fate of lifshits tails in magnetic fields

We investigate the integrated density of states of the Schrödinger operator in the Euclidean plane with a perpendicular constant magnetic field and a random potential. For a Poisson random potential with a nonnegative, algebraically decaying, single-impurity potential we prove that the leading asymptotic behavior for small energies is always given by the corresponding classical result, in contrast to the case of vanishing magnetic field. We also show that the integrated density of states of the operator restricted to the eingenspace of any Landau level exhibits the same behavior. For the lowest Landau level, this is in sharp contrast to the case of a Poisson random potential with a delta-function impurity potential.     

Density of states of random Schrödinger operators with a uniform magnetic field

We consider the density of states of Schrödinger operators with a uniform magnetic field and a random potential with a Gaussian distribution. We show that the restriction to the states of the first Landau level is equivalent to a scaling limit where one looks at the density of states near to the energy of the first Landau level and simultaneously lets the strength of the coupling to the random potential go to zero. We also consider a different limit where we look at the suitably normalised density of states near to the energy of the first Landau level when the intensity of the magnetic field goes to infinity.     

Magneto-oscillations in a two-dimensional electron gas: A simplified analytic formulation

We derive a simplified thermodynamical formulation for a two-dimensional electron gas (2DEG) in a uniform magnetic field, with a large Landau level width and at low temperatures. Our analytic results clearly bring out dependences of magneto-oscillations on the Landau level broadening.1. In contrast with Gaussian broadening, different Landau levels do not overlap in the case of semi-elliptic density of states (Ando and Uemura, 1974), were such a normalization constant needs not to be introduced.     

Landau levels line widths in potential scattering in the presence of a strong magnetic field

Summary The FBA scattering cross-section in the presence of a strong magnetic field diverges at the Landau thresholds. Such divergences are eliminated by the introduction of a modified density of states, accounting for the finite Landau states lifetime of the electrons in a magnetized plasma.     

Electronic properties of a Penrose tiling lattice in a magnetic field

The one-electron energy spectrum of a two-dimensional Penrose tiling lattice in a uniform magnetic field is calculated as a function of magnetic fields with a tight-binding Hamiltonian. The calculated results show the following remarkable features characteristic of the Penrose lattice. (1) The density of states in a magnetic field has a central peak with zero width at the zero energy. It is shown that the zero-energy states correspond to the ring states in which the wavefunction has a non-vanishing amplitudes only at the sites on a ring-like region around the origin. (2) The energy levels coalesce into Landau type levels and the boundary states due to the finite size effects based on a fixed boundary condition appear in the gap region between Landau levels. (3) The magnetic field dependence of the energy spectrum has a repeated pattern of self-similarity with the golden mean ratio of two successive periods.     

Landau levels in a simple hexagonal bilayer graphene

A new approach, which makes the Hamiltonian of the Peierls tight-binding model change into a band matrix, is used to investigate the Landau levels in a AA-stacked bilayer graphene. The interlayer atomic hoppings could induce an energy gap, the asymmetry of the Landau levels about the chemical potential, the random variation in the level spacing, more fourfold degenerate Landau levels at low energy, and the oscillatory Landau levels and the complicated state degeneracies at moderate energy. For the low-energy Landau levels, their dependence on the quantum number and the field strength cannot be well characterized by a simple power law. They exhibit a anomalous oscillation during the variation of the magnetic field. The main features of the magnetoelectronic states are directly reflected in density of states.     

Landau Hamiltonians with random potentials: Localization and the density of states

We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finitearea Green's functions. The proof of the decay estimates for the Green's functions uses fundamental results from two-dimensional bond percolation theory.Supported in part by CNRS.Supported in part by NSF grants INT 90-15895 and DMS 93-07438.Unité Propre de Recherche 7061.     

外场调控下硼磷烯量子点的电子结构和光学性质研究

采用紧束缚近似方法对锯齿状六边形硼磷烯量子点在平面电场和垂直磁场调控下的电子结构和光学性质进行了研究. 研究表明,硼磷烯量子点作为直接带隙半导体,在无外加电场和磁场作用时,能隙不随尺寸的改变而变化. 在平面电场调控下,能隙随电场强度的增加逐渐减小直至消失,平面电场方向几乎不会对硼磷烯量子点体系产生影响, 且随量子点尺寸的增大,能隙消失所需电场强度逐渐减小. 在垂直磁场调控下,表现为体态的能级在磁场作用下形成朗道能级,而能隙边缘处的朗道能级近似为一个平带,不随磁通量的改变而变化,态密度主要分布于朗道能级处. 另外,垂直磁场作用下的光吸收主要是由朗道能级之间的跃迁引起的.     

Point impurities remove degeneracy of the Landau levels in a two-dimensional electron gas

The density of states of a two-dimensional electron gas in a magnetic field has been studied taking into account the scattering on point impurities. It is demonstrated that allowance for the electron-impurity interaction completely removes degeneracy of the Landau levels even for a small volume density of these point defects. The density of states is calculated in a self-consistent approximation taking into account all diagrams without intersections of the impurity lines. The electron density of states ρ is determined by the contribution from a cut of the one-particle Green’s function rather than from a pole. In a wide range of the electron energies ω (measured from each Landau level), the value of ρ(ω) is inversely proportional to the energy |ω| and proportional to the impurity concentration. The obtained results are applicable to various two-dimensional electron systems such as inversion layers, heterostructures, and electrons on the surface of liquid helium.     

Tuning the effects of Landau level mixing on anisotropic transport in quantum Hall systems

Electron-electron interactions in half-filled high Landau levels in two-dimensional electron gases in a strong perpendicular magnetic field can lead to states with anisotropic longitudinal resistance. This longitudinal resistance is generally believed to arise from broken rotational invariance, which is indicated by charge density wave order in Hartree-Fock calculations. We use the Hartree-Fock approximation to study the influence of externally tuned Landau level mixing on the formation of interaction-induced states that break rotational invariance in two-dimensional electron and hole systems. We focus on the situation when there are two non-interacting states in the vicinity of the Fermi level and construct a Landau theory to study coupled charge density wave order that can occur as interactions are tuned and the filling or mixing are varied. We consider numerically a specific example where mixing is tuned externally through Rashba spin-orbit coupling. We calculate the phase diagram and find the possibility of ordering involving coupled striped or triangular charge density waves in the two levels. Our results may be relevant to recent transport experiments on quantum Hall nematics in which Landau level mixing plays an important role.     

Density of states of a two-dimensional electron in a strong magnetic field and a random potential

The density of states of a two-dimensional electron in a strong magnetic field moving in a periodic and a random potential is calculated. The results are compared with the density of states of the Landau model with disorder as obtained in the single band approximation. The limitations of the single band model are discussed.     

Electronic states on the surface of graphite

Graphite consists of graphene layers in an AB (Bernal) stacking arrangement. The introduction of defects can reduce the coupling between the top graphene layers and the bulk crystal producing new electronic states that reflect the degree of coupling. We employ low temperature high magnetic field scanning tunneling microscopy (STM) and spectroscopy (STS) to access these states and study their evolution with the degree of coupling. STS in magnetic field directly probes the dimensionality of electronic states. Thus two-dimensional states produce a discrete series of Landau levels while three-dimensional states form Landau bands providing a clear distinction between completely decoupled top layers and ones that are coupled to the substrate. We show that the completely decoupled layers are characterized by a single sequence of Landau levels with square-root dependence on field and level index indicative of massless Dirac fermions. In contrast weakly coupled bilayers produce special sequences reflecting the degree of coupling, and multilayers produce sequences reflecting the coexistence of massless and massive Dirac fermions. In addition we show that the graphite surface is soft and that an STM tip can be quite invasive when brought too close to the surface and that there is a characteristic tip-sample distance beyond which the effect of sample-tip interaction is negligible.     

Enhanced fluctuations of the tunneling density of states near the bottom of a landau band measured by a local spectrometer

We have found that the local density of state fluctuations (LDOSF) in a disordered metal, detected using an impurity in the barrier as a spectrometer, undergo enhanced (with respect to Shubnikov-de Haas and de Haas-van Alphen effects) oscillations in strong magnetic fields, omega(c)tau>/=1. We attribute this to the dominant role of the states near the bottom of Landau bands which give the major contribution to the LDOSF and are most strongly affected by disorder. We also demonstrate that in intermediate fields the LDOSF increase with field B in accordance with the results obtained in the diffusion approximation.     

The effect of exchange interaction on quasiparticle Landau levels in narrow-gap quantum well heterostructures

Using the 'screened' Hartree-Fock approximation based on the eight-band k·p Hamiltonian, we have extended our previous work (Krishtopenko et al 2011 J. Phys.: Condens. Matter 23 385601) on exchange enhancement of the g-factor in narrow-gap quantum well heterostructures by calculating the exchange renormalization of quasiparticle energies, the density of states at the Fermi level and the quasiparticle g-factor for different Landau levels overlapping. We demonstrate that exchange interaction yields more pronounced Zeeman splitting of the density of states at the Fermi level and leads to the appearance of peak-shaped features in the dependence of the Landau level energies on the magnetic field at integer filling factors. We also find that the quasiparticle g-factor does not reach the maximum value at odd filling factors in the presence of large overlapping of spin-split Landau levels. We advance an argument that the behavior of the quasiparticle g-factor in weak magnetic fields is defined by a random potential of impurities in narrow-gap heterostructures.     

Mesoscopic fluctuations of the Coulomb drag at nu = 1/2

We consider mesoscopic fluctuations of Coulomb drag transresistivity between two layers at a Landau level filling factor nu = 1/2 each. We find that, at low temperatures, sample to sample fluctuations exceed both the ensemble average and the corresponding fluctuations at B = 0. At the experimentally relevant temperatures, the variance of the transresistivity is proportional to T(-1/2). We find the dependence of this variance on density and magnetic field to reflect the attachment of two flux quanta to each electron.     

Condensation of states at the Landau levels and disorder-independent transport in a random system

I consider an electron on a square lattice, formed by random strenght point potentials, in the presence of a magnetic field, orthogonal to the lattice. When the number of magnetic flux quanta per unit cell is less than one, then an infinite density of extended disorder independent states condenses at each Landau level and provides the possibility for resistivity minima in a disordered system.     

Levitation of Extended States in a Random Magnetic Field with a Finite Mean

We study the localization properties of electrons in a two-dimensional system in a random magnetic field B(r)=B₀+δB(r) with the average B₀ and the amplitude of the magnetic field fluctuations δB. The localization length of the system is calculated by using the finite-size scaling method combined with the transfer-matrix technique. In the case of weak δB, we find that the random magnetic field system is equivalent to the integer quantum Hall effect system, namely, the energy band splits into a series of disorder broadened Landau bands, at the centers of which states are extended with the localization length exponent ν=2.34±0.02. With increasing δB, the extended states float up in energy, which is similar to the levitation scenario proposed for the integer quantum Hall effect.     

Lifshitz Tails in Constant Magnetic Fields

We consider the 2D Landau Hamiltonian H perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of H. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained by the first author and T. Wolff in [25] for the case of a vanishing magnetic field.     

On the Possibility of Superconductivity in Bilayer Heterostructures

A model is created for bilayer heterostructures in a strong magnetic field which makes it possible to neglect the Coulomb interaction. The thermodynamic instability of states of the electron system in a strong magnetic field leads to the formation of a periodic vortex lattice. The case is considered where the electron density is close to the density of the half-filled Landau level. An electron spectrum is found and an analog of the Cooper effect appearing under the Bogoliubov canonical transformation for electron Fermi operators is studied.