Boundary condition at the junction

Harmer, M.; Павлов, Б. С.; Yafyasov, A. M.

doi:10.1007/s10825-006-0085-7

Cited by 16 publications

(19 citation statements)

References 13 publications

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“…In what follows, we study second-order differential operators on metric graphs with matching conditions more general than those of δ−type, introduced above, namely, with the so-called weighted, or "Datta-Das Sarma", matching conditions, see [21,31,45]. In the case of differential expression (17) on the graph G ε , the corresponding modification is described as follows.…”

Section: Datta-das Sarma Conditionsmentioning

confidence: 99%

Norm-Resolvent Convergence of One-Dimensional High-Contrast Periodic Problems to a Kronig–Penney Dipole-Type Model

Cherednichenko

Kiselev

2016

Commun. Math. Phys.

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Abstract:We prove operator-norm resolvent convergence estimates for one-dimensional periodic differential operators with rapidly oscillating coefficients in the non-uniformly elliptic high-contrast setting, which has been out of reach of the existing homogenisation techniques. Our asymptotic analysis is based on a special representation of the resolvent of the operator in terms of the M-matrix of an associated boundary triple ("Krein resolvent formula"). The resulting asymptotic behaviour is shown to be described, up to a unitary transformation, by a non-standard version of the Kronig-Penney model on R.

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Section: Datta-das Sarma Conditionsmentioning

confidence: 99%

Norm-Resolvent Convergence of One-Dimensional High-Contrast Periodic Problems to a Kronig–Penney Dipole-Type Model

Cherednichenko

Kiselev

2016

Commun. Math. Phys.

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“…The quantum-mechanical meaning of the similar parameter β in the case of T-junction is revealed in [12]. In our case, the parameter is defined by the geometry of the contact zone.…”

Section: The Resonance Interpretationmentioning

confidence: 85%

“…In [7] we base our conclusions upon a simpler model, substituting inner structure by a finite matrix, which is fit based on experimental data on size-quantization. Similarly to [2] we emulate the barrier in [7] by the generalized Datta and Das Sarma boundary condition, see [7,12,13]. The 1D solvable model of the contact zone of the emitter is constructed hereafter based on division of the normal coordinate into three layers; the metal (1) ∞ < x < 0, the vacuum (2), a < x < ∞ and carbon deposit (3), 0 ≤ x ≤ a., with the portions of the wave-function denoted correspondingly.…”

Section: The Resonance Interpretationmentioning

confidence: 99%

“…In particular the sum is vanishing while the boundary data are imposed to the generalized Datta-Das Sarma boundary condition at the contact of the deposit and vacuum (that is exactly on the barrier). This boundary condition is defined, similarly to Datta-Das Sarma, [12], by the vector parameter β = (β 1 , β 2 , β 3 ) as:…”

Section: The Resonance Interpretationmentioning

confidence: 99%

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Dimensional Quantization and the Resonance Concept of the Low-Threshold Field Emission

et al. 2015

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We present a brief critical review of modern theoretical interpretations of the low-threshold field emission phenomenon for metallic electrodes covered with carbon structures, taking the latest experiments into consideration, and confirming the continuity of spectrum of resonance states localized on the interface of the metallic body of the cathode and the carbon cover. Our proposal allowed us to interpret the double maxima of the emitted electron's distribution on full energy. The theoretical interpretation is presented in a previous paper which describes the (1 + 1) model of a periodic 1D continuous interface. The overlapping of the double maxima may be interpreted taking into account a 2D superlattice periodic structure of the metal-vacuum interface, while the energy of emitted electrons lies on the overlapping spectral gaps of the interface 2D periodic lattice.

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“…The classical Fowler-Nordheim techniques for calculating the transmission coefficient T for simple rectangular potential barrier, see [8], gives an exponentially small value T ≈ e −qa with q = v − 2mEh −2 for the under-barrier tunneling with v >> 2mEh −2 and the width of the barrier equal to a. The resonance modification of the classical Fowler-Nordheim algorithm for calculating the transmission coefficient across a rectangular barrier, in presence of the energy levels of the size-quantization, meets some technical complications which can be avoided while substituting the rectangular barrier with delta-barrier supplied with an inner structure, attached to the barrier by Datta and Das-Sarma boundary condition, see [9,10] for discussion of this phenomenological boundary condition and the derivation of it from the first principles in [11]. The program of resonance interpretation of the low-threshold field emission, based on zero-range model barrier with an inner structure, is developed in [7].…”

Section: An Example: Resonance Concepts Of the Low-threshold Field Emmentioning

confidence: 99%

Resonance scattering across the superlattice barrier and the dimensional quantization

Павлов¹,

Yafyasov²

2016

Nanosystems: Phys. Chem. Math.

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Carbon nano-cluster cathodes exhibit a low threshold electron emission, which is 2-3 orders lower than on metals and semiconductors. We confirm the effect by direct experiments with graphene structures. We are suggesting a model based on the interference electrons wave function in 3D-space charge region of carbon structure interface with vacuum. The low-threshold emission is explained, in frames of the model, by the resonance properties of the barrier formed on the interface. Also in the following topics: interpretation of recent experimental findings for saturation of the field emission; local spectral analysis of multidimensional periodic lattices: dispersion via DN-map; examples of iso-energetic surfaces associated with solvable models of periodic lattice; Lagrangian version of the operator extension algorithm; solvable models of selected one-body spectral problems; quantum dot attached to the node of a quantum graph; a solvable model of a discrete lattice and spectral structure of a 1D superlattice via analytic perturbation procedure.Keywords: periodic interface, carbon cover, Weyl-Titchmarsh function, resonance, field emission, DN-map. Received: 23 August 2016Revised: 1 September 2016 An example: resonance concepts of the low-threshold field emissionIn numerous recent experiments, see for instance [1][2][3][4][5][6][7] extremely low-threshold field emission from metallic cathodes under carbon deposit was observed for electric fields (10 4 − 10 5 V /cm). This is a surprisingly strong effect, because the field initiating a noticeable emission (10 −10 − 10 −9 A) from these materials is by 2-3 orders of magnitude less than the field required for the field-emission from the traditional metals and semiconductors. Despite an obviously unusual nature of the effect, numerous authors, see for instance [5,6] attempted to explain the low-threshold phenomenon trivially with use of the classical Fowler-Nordheim techniques, based on enhancing of the field at the micro-protrusions. They assume that the local field F s near the emitting center is calculated as F 0 = γF 0 , where γ is the field enhancement coefficient, defined by the micro-geometry, and F 0 is the field of the equivalent flat capacitor. This completely classical explanation of the low-threshold emission phenomenon is not universal, and certainly non-valid for deposit, considered in our recent papers [3,7] because the surface of the carbon flakes, obtained by the detonation synthesis, are perfectly smooth, see the flakes (see Fig. 1) under maximal magnifications.with rare and relatively small protrusions. These protrusions are able to lower the threshold 5-fold, while 10 2 times lowering is observed in our experiments. We suggested in in [3,7] an alternative explanation of the threshold lowering (field enhancement) based on the dimensional (size-) quantization in the under-surface spacecharge region. The classical Fowler-Nordheim techniques for calculating the transmission coefficient T for simple rectangular potential barrier, see [8], gives an exponentially...

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Boundary condition at the junction

Abstract: The quantum graph plays the role of a solvable model for a two-dimensional network. Here fitting parameters of the quantum graph for modelling the junction is discussed, using previous results of the second author.Comment: Replaces unpublished draft on related researc

Cited by 16 publications

References 13 publications

Norm-Resolvent Convergence of One-Dimensional High-Contrast Periodic Problems to a Kronig–Penney Dipole-Type Model

Norm-Resolvent Convergence of One-Dimensional High-Contrast Periodic Problems to a Kronig–Penney Dipole-Type Model

Dimensional Quantization and the Resonance Concept of the Low-Threshold Field Emission

Resonance scattering across the superlattice barrier and the dimensional quantization

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