321 research outputs found
Symmetric integrators with improved uniform error bounds and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime
In this paper, we are concerned with symmetric integrators for the nonlinear
relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter
, which is inversely proportional to the speed of light.
The highly oscillatory property in time of this model corresponds to the
parameter and the equation has strong nonlinearity when \eps is
small. There two aspects bring significantly numerical burdens in designing
numerical methods. We propose and analyze a novel class of symmetric
integrators which is based on some formulation approaches to the problem,
Fourier pseudo-spectral method and exponential integrators. Two practical
integrators up to order four are constructed by using the proposed symmetric
property and stiff order conditions of implicit exponential integrators. The
convergence of the obtained integrators is rigorously studied, and it is shown
that the accuracy in time is improved to be \mathcal{O}(\varepsilon^{3}
\hh^2) and \mathcal{O}(\varepsilon^{4} \hh^4) for the time stepsize \hh.
The near energy conservation over long times is established for the multi-stage
integrators by using modulated Fourier expansions. These theoretical results
are achievable even if large stepsizes are utilized in the schemes. Numerical
results on a NRKG equation show that the proposed integrators have improved
uniform error bounds, excellent long time energy conservation and competitive
efficiency
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Spectroscopic Study of Localized States in Twisted Semiconducting Heterostructures and Charge Transfer Driven Phenomena in a-RuCl₃ Heterointerfaces
This thesis investigates the unique properties of 2D devices such as twisted semiconducting bilayers and a-RuCl₃ heterostructures employing scanning tunneling microscopy (STM) and spectroscopy (STS) probes. The research presented here sheds light on the vast opportunities that 2D materials provide in condensed matter systems as well as future device applications. Among 2D materials, transition metal dichalcogenide (TMD) heterobilayers provide a promising platform to study many quantum phenomena such as excitonic states due to their tunability of band gap. In addition, TMDs are excellent candidates to achieve localized states and carrier confinement, crucial for single photon emitters used in quantum computation and information. We begin this thesis with a brief overview of STM/STS and utilizing these techniques on 2D materials in the first and second chapters.The third chapter of this work investigates the twisted bilayer of WSe₂ and MoSe₂ in the H-stacking configuration using STM/STS which was previously challenging to measure. The spectroscopic results obtained from the heterobilayer indicate that a combination of structural rippling and electronic coupling generates unexpectedly large \moire potentials, in the range of several hundred meV. Our analysis reveals that the \moire structure and internal strain, rather than interlayer coupling, are the main factors of the moire potential. Large moire potentials lead to deeply trapped carriers such as electron-hole pairs, so-called excitons. Our findings open new routes toward investigating excitonic states in twisted TMDs.
In the next chapter, we investigate the ultralocalized states of twisted WSe₂/MoSe₂ nanobubbles. Mechanical and electrical nanostructurings are expected to modify the band properties of transition metal dichalcogenides at the nanoscale. To visualize this effect, we use STM and near-field photoluminescence to examine the electronic and optical properties of nanobubbles in the semiconducting heterostructures. Our findings reveal a significant change in the local bandgap at the nanobubble, with a continuous evolution towards the edge of the bubble. Moreover, at the edge of the nanobubble, we show the formation of in gap bound states. A continuous redshift of the interlayer exciton on entering the bubble is also detected by the nano-PL. Using self-consistent Schrodinger-Poisson simulations, we further show that strong doping in the bubble region leading to band bending is responsible for achieving ultralocalized states. Overall, this work demonstrates the potential of 2D TMDs for developing well-controlled optical emitters for quantum technologies and photonics.
We next turn to the effect of the electric field in band gap tuning of WSe₂/WS₂ heterobilayer. The tunability of band gap is a crucial element in device engineering to achieve quantum emitters. The electrostatic gate generates doping and an electric field giving access to continuous tunability, higher doping level, and integration capability to nanoelectronic devices. We employ scanning tunneling microscopy (STM) and spectroscopy (STS) to probe the band properties of twisted heterobilayer with high energy and spatial resolution. We observe continuous band gap tuning up to several hundreds of meV change by sweeping the back gate. We introduced a capacitance model to take into account the finite tip size leading to an enhanced electric field. The result of our calculation captures well the band gap change observed by STS measurements. Our study offers a new route toward creating highly tunable semiconductors for carrier confinement in quantum technology.
In the next chapters, we focus on a-RuCl₃ heterointerfaces. We first explore the nanobubble of graphene/a-RuCl3 to create sharp p-n junctions. The ability to create sharp lateral p-n junctions is a critical requirement for the observation of numerous quantum phenomena. To accomplish this, we used a charge-transfer based heterostructure consisting of graphene and a-RuCl₃ to create nanoscale lateral p-n junctions in the vicinity of nanobubbles. Our approach relied on a combination of scanning tunneling microscopy (STM) and spectroscopy (STS), as well as scattering-type scanning near-field optical microscopy (s-SNOM), which allowed us to examine both the electronic and optical responses of these nanobubble p-n junctions. Our results showed a massive doping variation across the nanobubble with a band offset of 0.6 eV. Further, we observe the formation of an abrupt junction along nanobubble boundaries with an exceptionally sharp lateral width (<3 nm). This is one order of magnitude smaller length scale than previous lithographic methods. Our work paves the way toward device engineering via interfacial charge transfer in graphene and other low-density 2D materials.
In chapter 7, we describe the use of low-temperature scanning tunneling microscopy (STM) measurements to observe the \moire pattern in graphene/a-RuCl3 heterostructure to validate the InterMatch method. This method is effective in predicting the charge transfer, strain, and stability of an interface. The InterMatch method was applied to moire patterns of graphene/a-RuCl3 to predict the stable interface structure. STM topographs show three regions with distinct moire wavelengths due to atomic reconstructions. Using the InterMatch method, we perform a comprehensive mapping of the space of superlattice configurations and we identify the energetically favorable superlattices that occur in a small range of twist angles. This range is consistent with the STM results. Moreover, the spectra on these regions exhibit strong resonances with the spacing between resonances following the expectation from Landau levels on a Dirac spectrum due to strain and doping. The results of our scanning tunneling microscopy (STM) measurements confirm that the InterMatch method is effective in predicting the charge transfer and stability of interfaces between materials.
We next investigate WSe₂/a-RuCl₃ heterostructure through a multi-faceted approach. Our exploration encompassed diverse techniques such as STM, and optical measurements. We detect a significant charge transfer between the two layers by STM measurements, leading to a shift in the Fermi level towards the valence band of WSe₂. Our findings are supported by optical measurements and DFT calculations, which confirm the p-doped WSe₂ observed through STM. The results of this work highlight a-RuCl₃ potential for contact engineering of TMDs and unlocking their functionalities for the next generation optoelectronic devices.
In the last chapter of this thesis, I provide a brief conclusion as well as a few future directions and insights for investigating 2D materials
Efficient computation of the sinc matrix function for the integration of second-order differential equations
This work deals with the numerical solution of systems of oscillatory
second-order differential equations which often arise from the
semi-discretization in space of partial differential equations. Since these
differential equations exhibit pronounced or highly) oscillatory behavior,
standard numerical methods are known to perform poorly. Our approach consists
in directly discretizing the problem by means of Gautschi-type integrators
based on matrix functions. The novelty contained here is
that of using a suitable rational approximation formula for the
matrix function to apply a rational Krylov-like
approximation method with suitable choices of poles. In particular, we discuss
the application of the whole strategy to a finite element discretization of the
wave equation
Exponential integrators: tensor structured problems and applications
The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed
Product integration rules by the constrained mock-Chebyshev least squares operator
In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of "pathological" behavior, e.g. "nearly" singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates
Au delà de la dissipation markovienne à l’échelle nanométrique : vers la découverte de règles quantiques pour la conception de nano-dispositifs bio-organiques
A better understanding of dissipation is crucial for understanding real-world quantum systems. Indeed, all quantum systems experience interactions with an (often) uncontrollable outside environment that can lead to a decay of excited state populations and a loss of quantum coherences. The study of dissipation is timely as the development of next-generation nanoscale quantum technologies is on its way, and the existence of non-trivial quantum effects in biological systems is being seriously investigated. However, descriptions of dissipation in quantum systems are reduced (most of the time) to time-local approaches and (everywhere) to space-local independent environments. These simplifying assumptions do render analytic and numerical calculations possible, yet they get rid of a breadth of physical processes that can alter radically the quantum systems' dynamics. In this thesis, building on a numerically exact tensor networks method, we developed a technique able to handle spatio-temporal correlations between a quantum system and bosonic (i.e. vibrational, electromagnetic, magnons, etc.) environments. With this method we studied the signalling process - a form of information backflow - in quantum systems, and uncovered how it can induce non-trivial dynamics, and be leveraged to populate otherwise inaccessible excited states. We also evidenced the ability of 'non-local' environment reorganisation, induced by system-environment interactions, to radically change the nature of the thermodynamically favoured system ground state. The new phenomenology of physical processes, resulting from considering quantum systems interacting with a common environment, has important consequences for the design of nanodevices as it gives access to new control, sensing and cross-talk mechanisms. In another vein, these results might also give us a new framework to study and interpret (quantum?) effects in the biological realm
A Lotka-Volterra type model analyzed through different techniques
We consider a modified Lotka-Volterra model applied to the predator-prey
system that can also be applied to other areas, for instance the bank system.
We show that the model is well-posed (non-negativity of solutions and
conservation law) and study the local stability using different methods.
Firstly we consider the continuous model, after which the numerical schemes of
Euler and Mickens are investigated. Finally, the model is described using
Caputo fractional derivatives. For the fractional model, besides well-posedness
and local stability, we prove the existence and uniqueness of solution.
Throughout the work we compare the results graphically and present our
conclusions. To represent graphically the solutions of the fractional model we
use the modified trapezoidal method that involves the modified Euler method.Comment: Accepted on June 22, 2023 for publicatio
Is polynomial interpolation in the monomial basis unstable?
In this paper, we show that the monomial basis is generally as good as a
well-conditioned polynomial basis for interpolation, provided that the
condition number of the Vandermonde matrix is smaller than the reciprocal of
machine epsilon. We also show that the monomial basis is more advantageous than
other polynomial bases in a number of applications.Comment: 30 pages, 12 figure
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