Shifted Landau levels in curved graphene sheets

Debus, Jens-Daniel; Mendoza, M.; Herrmann, Hans J.

doi:10.1088/1361-648x/aadecd

Cited by 16 publications

(24 citation statements)

References 29 publications

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“…Charge carriers in graphene can be described theoretically by a 2-D Dirac equation in curved space-time, where the metric is related to the graphene sample deformation [45]. The dynamics of these charge carriers was recently studied numerically [51,52] with a lattice Boltzmann method. Using our numerical schemes, we now propose some tests with similar configurations.…”

Section: Massless Dirac Particles In Curved Graphenementioning

confidence: 99%

“…One of the goals of this article is to extend this numerical scheme to the Dirac equation in curved space.Recently, the Dirac equation in curved space-time has gained important interest in some applications such as condensed matter physics for describing the dynamics of charge carriers in deformed 2D Dirac materials [44,45], as well as astrophysics for fermion tunneling in black holes [46,47,48,49]. In its discrete version, it has been considered from the lattice Boltzmann technique point of view [50,51,52] and as continuous limits of quantum walks [53,54,55]. However, the literature on numerical methods for this equation is scarse.…”

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confidence: 99%

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Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Antoine

Fillion‐Gourdeau²,

Lorin³

et al. 2020

Journal of Computational Physics

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Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of non-constant coefficients in the curved space Dirac equation, convolution products usually appear when the Fourier transform is performed. The strategy based on pseudodifferential operators allows for efficient computations of these convolution products by using an ordinary fast Fourier transform algorithm. The resulting numerical methods are efficient and have spectral convergence. Simultaneously, wave absorption at the boundary can be easily derived using absorbing layers to cope with some potential negative effects of periodic conditions inherent to spectral methods. The numerical schemes are first tested on simple systems to verify the convergence and are then applied to the dynamics of charge carriers in strained graphene. 12], tremendous efforts have been put these past two decades on the development of numerical methods for the computation of the time-dependent Dirac equation in flat Minkowski space. Real space methods such as Quantum Lattice Boltzman techniques [13,14,15,16,17], Galerkin methods [18,19,20], and pseudospectral or spectral methods [21,22,23,24,25,26,27,28] were developed to efficiently and accurately solve the Dirac equation. Semi-classical regimes were considered in [29] using Gaussian beams or Frozen Gaussian Approximations [30,31], while the non-relativistic limit has been studied in several recent papers [32,33]. The computational difficulties for solving the time-dependent Dirac equation include the fermion doubling problem, related to numerical dispersion [16], and the Zitterbewegung [1], resulting in highly oscillating solutions whose origin can be traced back to the presence of the mass term βmc 2 . Finally, drastic stability conditions can lead to numerical diffusion related to the finite wave propagation speed, the speed of light c.Another numerical challenge for the Dirac equation shared by any other wave equation in real space solved on a truncated domain is the need of imposing special boundary conditions in order to avoid spurious wave reflections at the computational domain boundary. Therefore, the computational methods on truncated domains require non-reflective boundary conditions [15,34,35,36,37], absorbing or perfectly matched layers (PML) [35,38,39,40,41], or the introduction of an artificial potential [38]. On the other hand, Fourier-based methods applied on bounded domains naturally induce periodic boundary conditions, which can be problematic when dealing with delocalized wave functions. In this respect, the technique developed in [42,43] for the Dirac equation in flat space, where a spectral method is combined with PML, is an interesting alternative. One of the goals of this article is to extend this numerical scheme to the Dirac equation in curved space.Recently, the Dirac equation in curved space-time has gained important intere...

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Section: Massless Dirac Particles In Curved Graphenementioning

confidence: 99%

mentioning

confidence: 99%

Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Antoine

Fillion‐Gourdeau²,

Lorin³

et al. 2020

Journal of Computational Physics

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“…The numerical solutions are obtained with the Quantum Lattice Boltzmann Method as described in Ref. [11].…”

Section: Theory Of Strained Graphenementioning

confidence: 99%

“…The energy spectrum is obtained from the time dependence of the wave-function via the Fourier transform, Eq. (11). The Landau levels are seen as the discrete peaks corresponding to n = 1, 3, 5.... Only the odd energy states are excited due to the wave-function being symmetric and no anti-symmetric eigenstates are initialized.…”

Section: A Shifted Landau Levels In Curved Graphene Sheetsmentioning

confidence: 99%

“…The method is extended to curved spaces [14,15] and related to strained graphene by exploiting the similarities between the curved Dirac Hamiltonian and the effective Hamiltonian for strained graphene as done in Ref. [11]. This opens up the possibility of exporting the proven advantages of the curved space lattice Boltzmann meth-ods, such as computational efficiency, parallelizability and easy handling of complex geometries, to graphene.…”

Section: Introductionmentioning

confidence: 99%

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Landau levels in wrinkled and rippled graphene sheets

Flouris

Mendoza

Herrmann

2019

Int. J. Mod. Phys. C

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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 average space curvatures and shown to obey a linear relation. The study is carried out with a quantum lattice Boltzmann method, solving the Dirac equation on curved manifolds.

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Gravitational Waves and Helicity in Undoped Graphene

Sorge

2020

Annalen der Physik

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The behavior of a monolayer undoped graphene sheet in presence of a weak gravitational wave is discussed. In the low‐energy limit, the linear dispersion of the energy levels near the Dirac points can be ascribed to massless excitations obeying a Dirac equation in a 2+1D spacetime. Starting from a 2+1 dimensional (2+1D) Lagrangian, the equations of motion for the electron (hole) wavefunction in the graphene sheet are deduced and solved. The Bogolubov coefficients are then obtained and the gravitationally induced amplitude of the electron–hole pair excitations is evaluated. The typical quadrupolar distribution of the pair creation probability appears rotated by a angle in the plane of the graphene sheet, when compared to the case of a confined 2+1D massless scalar field in the same gravitational background. Such behavior can be understood taking into account the role of helicity in graphene during the interaction with the gravitational wave. Although very small, the effect offers an interesting example of interaction at nanoscale between a non‐trivial quantum system and a time‐dependent gravitational background. The strong analogy between the quantum theoretical description of the graphene physics and the standard quantum field theory can also suggest a similar analysis in different physical contexts.

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Shifted Landau levels in curved graphene sheets

Cited by 16 publications

References 29 publications

Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Landau levels in wrinkled and rippled graphene sheets

Gravitational Waves and Helicity in Undoped Graphene

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