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.

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
• 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: https://theses.gla.ac.uk/id/eprint/5333