Band-structure and electronic transport calculations in cylindrical wires: the issue of bound states in transfer-matrix calculations

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Résumé

The transfer-matrix methodology is used to solve linear systems of differential equations, such as those that arise when solving Schrödinger's equation, in situations where the solutions of interest are in the continuous part of the energy spectrum. The technique is actually a generalization in three dimensions of methods used to obtain scattering solutions in one dimension. Using the layer-addition algorithm allows one to control the stability of the computation and to describe efficiently periodic repetitions of a basic unit. This paper, which is an update of an article originally published in Physical and Chemical News 16, 46-53 (2004), provides a pedagogical presentation of this technique. It describes in details how the band structure associated with an infinite periodic medium can be extracted from the transfer matrices that characterize a single basic unit. The
method is applied to the calculation of the transmission and band structure of electrons subject to cosine potentials in a cylindrical wire. The simulations show that bound states must be considered because of their impact as sharp resonances in the transmission probabilities and to remove unphysical discontinuities in the band structure. Additional states only improve the completeness of the representation.
langue originaleAnglais
Numéro d'article1907.06940
Nombre de pages13
journalArXiv pre-print
étatPublié - 16 juil. 2019

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wire
news
completeness
linear systems
electronics
repetition
discontinuity
energy spectra
differential equations
methodology
scattering
electrons
simulation

mots-clés

  • transfer matrix
  • bound states
  • band structure
  • electronic transport

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title = "Band-structure and electronic transport calculations in cylindrical wires: the issue of bound states in transfer-matrix calculations",
abstract = "The transfer-matrix methodology is used to solve linear systems of differential equations, such as those thatarise when solving Schr ̈odinger’s equation, in situations where the solutions of interest are in the continuouspart of the energy spectrum. The technique is actually a generalization in three dimensions of methods usedto obtain scattering solutions in one dimension. Using the layer-addition algorithm allows one to control thestability of the computation and to describe efficiently periodic repetitions of a basic unit. This paper, whichis an update of an article originally published in Physical and Chemical News 16, 46-53 (2004), provides apedagogical presentation of this technique. It describes in details how the band structure associated with aninfinite periodic medium can be extracted from the transfer matrices that characterize a single basic unit.The method is applied to the calculation of the transmission and band structure of electrons subject to cosinepotentials in a cylindrical wire. The simulations show that bound states must be considered because of theirimpact as sharp resonances in the transmission probabilities and to remove unphysical discontinuities in theband structure. Additional states only improve the completeness of the representation",
keywords = "transfer matrix, bound states, band structure, electronic transport, electronic transport, transfert matrix methodology, S matrices, band structure calculation, bound states, quantum wires",
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AB - The transfer-matrix methodology is used to solve linear systems of differential equations, such as those thatarise when solving Schr ̈odinger’s equation, in situations where the solutions of interest are in the continuouspart of the energy spectrum. The technique is actually a generalization in three dimensions of methods usedto obtain scattering solutions in one dimension. Using the layer-addition algorithm allows one to control thestability of the computation and to describe efficiently periodic repetitions of a basic unit. This paper, whichis an update of an article originally published in Physical and Chemical News 16, 46-53 (2004), provides apedagogical presentation of this technique. It describes in details how the band structure associated with aninfinite periodic medium can be extracted from the transfer matrices that characterize a single basic unit.The method is applied to the calculation of the transmission and band structure of electrons subject to cosinepotentials in a cylindrical wire. The simulations show that bound states must be considered because of theirimpact as sharp resonances in the transmission probabilities and to remove unphysical discontinuities in theband structure. Additional states only improve the completeness of the representation

KW - transfer matrix

KW - bound states

KW - band structure

KW - electronic transport

KW - electronic transport

KW - transfert matrix methodology

KW - S matrices

KW - band structure calculation

KW - bound states

KW - quantum wires

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JO - ArXiv pre-print

JF - ArXiv pre-print

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