Energy bands and forbidden gaps in the Kronig-Penney model

Folland, N. O.

doi:10.1103/physrevb.28.6068

Cited by 9 publications

(4 citation statements)

References 11 publications

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“…The neutron band structure with a square well potential (whose analytic solution was found a long time ago by Kronig and Penney [18,19]) was the only case discussed by Oyamatsu and Yamada [2]. The single particle wavefunction can be factored in the form 10) in which the reduced wave funtion φ k z is the solution of a one dimensional equation 11) satisfying the boundary conditions…”

Section: "Lasagna" Phasementioning

confidence: 99%

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Entrainment coefficient and effective mass for conduction neutrons in neutron star crust: simple microscopic models

Carter¹,

Chamel²,

Haensel³

2005

Nuclear Physics A

136

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Abstract. In the inner crust of a neutron star, at densities above the "drip" threshold, unbound "conduction" neutrons can move freely past through the ionic lattice formed by the nuclei. The relative current density n i = nv i of such conduction neutrons will be related to the corresponding mean particle momentum p i by a proportionality relation of the form n i = Kp i in terms of a physically well defined mobility coefficient K whose value in this context has not been calculated before. Using methods from ordinary solid state and nuclear physics, a simple quantum mechanical treatment based on the independent particle approximation, is used here to formulate K as the phase space integral of the relevant group velocity over the neutron Fermi surface. The result can be described as an "entrainment" that changes the ordinary neutron mass m to a macroscopic effective mass per neutron that will be given -subject to adoption of a convention specifying the precise number density n of the neutrons that are considered to be "free" -by m ⋆ = n/K . The numerical evaluation of the mobility coefficient is carried out for nuclear configurations of the "lasagna" and "spaghetti" type that may be relevant at the base of the crust. Extrapolation to the middle layers of the inner crust leads to the unexpected prediction that m ⋆ will become very large compared with m .

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Section: "Lasagna" Phasementioning

confidence: 99%

Section: "Lasagna" Phasementioning

confidence: 99%

Entrainment coefficient and effective mass for conduction neutrons in neutron star crust: simple microscopic models

Carter¹,

Chamel²,

Haensel³

2005

Nuclear Physics A

136

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“…Thus, contrary to a number of misleading remarks in the literature, the nearly-free electron model is not applicable and the width of the mini-bands is best estimated within a tight-binding model or by implementing periodic boundary conditions and matching the envelope functions from the adjacent layers (e.g. Bastard 1981) (for a more advanced version of the Kronig-Penney model see Folland (1983)). The degree of overlap, the corresponding band widths and tunnelling parameters (e.g.…”

Section: F ' ( X ) U ~ ~( ~)mentioning

confidence: 99%

Electronic properties of semiconductor alloy systems

Jaroš¹

1985

Rep. Prog. Phys.

176

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Semiconductor alloys provide a natural means of tuning the magnitude of the forbidden gap and other material parameters so as to optimise and widen the application of semiconductor devices. With the advent of small-structure systems, such as quantum wells and superlattices, the effects of alloy composition, size, device geometry, doping and controlled lattice strain can be combined to achieve maximum tunability. In this article, the concepts and ideas peculiar to the description and modelling of semiconductor alloy systems are reviewed with a view to providing a link between electronic structure and optical and transport properties.The material parameters of intrinsic and extrinsic alloys are normally derived from those of the endpoint compounds in terms of a simple interpolation procedure. This model assumes that the material is a nearly perfect random alloy and that the electronic structure can be generated from suitably averaged parameters of the constituent compounds. Any deviation from this virtual crystal approximation (VCA) is treated separately as a perturbation. Consequently, the concepts and methods developed for compound semiconductors can be employed in a straightforward manner to describe alloys. However, recent experimental and theoretical studies on several semiconductor alloys (e.g. GaInAs, HgCdTe) indicate that the VCA breaks down whenever the mismatch between the electronic properties of the constituent atoms exceeds a certain critical value. The breakdown of the VCA is manifested by (i) the formation of ordered structures of lower symmetry, and (ii) the weakening of sp3 bonds on one sublattice which leads to selective localisation and charge transfer. In the case of defects, the position of deep levels and resonances reflects strong mixing by the short-range part of the potential of Bloch states spanning a range of many tens of volts. The relationship between the level depth and localisation as a function of alloy composition is non-linear and unrelated to the bowing of the alloy band gap. In the assessment of the electronic properties of submicron structures containing an alloy semiconductor, the bulk alloy properties and the effects of confinement, doping and strain must be treated on an equal footing. The bulk alloy states are mixed and the corresponding optical and transport parameters are changed. A certain degree of such mixing normally occurs in lattice-matched undoped systems as a result of the interaction between the states belonging to adjacent layers. This leads to the formation of well localised states which 0034-4885/85/081091+ 64$09.00 @ 1985 The Institute of Physics 10911092 M Jaros lie outside the confining barriers and which cannot be properly accounted for by the simple envelope function model. In narrow and medium-thick wells (< 100 A) the confinement effect is a sensitive function of the short-wavelength Fourier components of the potential. A breakdown of the vcA-which alters the description of alloy properties in a qualitative manner-has significant consequences for confine...

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“…Different from the traditional algebra evaluation methodologies, the equivalent circuit method of the Kronig-Penney model can easily show the presence of the allowed and the forbidden bands and E versus k relation only by using some quantum wells. The energy band concept is the energy levels split for many quantum wells and the Kronig-Penney model is to describe the band behaviour of periodic potential of actual crystal [14]. The Figure 1.…”

Section: The Kronig-penney Model Simulationmentioning

confidence: 99%

Kronig–Penney model simulation with equivalent circuit method

Tsai

2006

Int J Numerical Modelling

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SUMMARYThe Kronig-Penney model is quite useful for illustrating many important features of the quantum behaviour of electrons in periodic lattice. Although the Kronig-Penney model is well-known and has been discussed in solid state textbooks, we try to use a simple and accessible way without the extremely laborious and tedious algebra evaluation to solve Kronig-Penney model. This paper presents a simple method without solving the difficult eigen-problem to solve the Kronig-Penney model, and the important energy band characteristics can be easily obtained with circuit concepts. The simulation results are presented to demonstrate the accuracy and superiority of the model.

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Energy bands and forbidden gaps in the Kronig-Penney model

Cited by 9 publications

References 11 publications

Entrainment coefficient and effective mass for conduction neutrons in neutron star crust: simple microscopic models

Entrainment coefficient and effective mass for conduction neutrons in neutron star crust: simple microscopic models

Electronic properties of semiconductor alloy systems

Kronig–Penney model simulation with equivalent circuit method

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