A novel network representation for modelling the electronic wavefunction in two dimensional quantum systems

Pepin, Jeremy (1990) A novel network representation for modelling the electronic wavefunction in two dimensional quantum systems. PhD thesis, University of Glasgow.

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Abstract

An overview of quantum phenomena associated with nanoelectronic structures is presented, including resonant tunnelling and mini-band formation in vertical transport devices and channel conductance quantization and interference in lateral devices. The method of construction of these structures is briefly described. Methods of calculating the transmission coefficient are reviewed. In one dimension the transfer matrix method is described and also two derivatives of the approach for circumventing the numerical instability encountered when calculating the wavefunction. In two dimensions an un-coupled matching states method and an asymptotic time dependent method are described. As an alternative to the above methods a coupled network theory is presented for the first time which genuinely represents the 2D time independent electronic wavefunction. Nodes on the network are described by a unitary scattering matrix from which a 2D transfer matrix is derived, connecting lines on the network. The scattering matrix for the whole system is created by combining the 2D scattering matrices for each line, themselves derived from the transfer matrices. The use of the scattering matrix is necessary to ensure numerical stability and current conservation. It is shown that the bandstructure of the network is essential to creating a genuine 2D model whilst at the same time introducing a perturbing influence on the manifestation of physical phenomena. The advantages over other models is the complete absence of restriction on the potential profile considered and no requirement to separate the scalar energy and potential quantities into x and y components. Also no problem with current continuity has been encountered. A major disadvantage is the large time required to calculate wavefunctions compared with the un-coupled matching states method. The network is shown to reproduce the channel conductance quantization recently observed experimentally and is in good agreement with both a 1D analytic model and a 2D un-coupled model. The network is applied to channels containing single and double barriers. In the latter case the resonances are found not to coincide with those predicted by a 1D model. Also the wavefunction on resonance resembles one of the quasi states of the well but with a phase shift. When applied to waveguides involving an interface between channels of different widths the network reveals a tendency for the wavefunction to relax to its original transverse state as it gets further from the interface. This tendency is most pronounced for a tapered junction at low energy (energy of the order of the first transverse eigenvalue). The transmission coefficient for an abrupt junction displays unusual dips above the quantization threshold of the narrow channel. Scattering into higher modes is reduced both by reducing the ratio of channel widths and by reducing the absolute lengths of the device. Finally circle and ring devices are studied, results displaying similarities with Finch's time dependent calculations. In particular scattering into the arms of the ring is observed to be mainly into the first mode if the energy is low and mainly into the third mode if the energy is of the same order as the third transverse eigenvalue of the channel. The tendency to relax into the original transverse state still operates over the whole device.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Subjects: T Technology > TK Electrical engineering. Electronics Nuclear engineering
Colleges/Schools: College of Science and Engineering > School of Engineering > Electronics and Nanoscale Engineering
Funder's Name: UNSPECIFIED
Supervisor's Name: Barker, Professor John
Date of Award: 1990
Depositing User: Ms Dawn Pike
Unique ID: glathesis:1990-5333
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 01 Jul 2014 09:03
Last Modified: 01 Jul 2014 09:03
URI: http://theses.gla.ac.uk/id/eprint/5333

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