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

[摘要] An overview of quantum phenomena associated with nanoelectronic structures ispresented, including resonant tunnelling and mini-band formation in verticaltransport devices and channel conductance quantization and interference in lateraldevices. The method of construction of these structures is briefly described.Methods of calculating the transmission coefficient are reviewed. In onedimension the transfer matrix method is described and also two derivatives of theapproach for circumventing the numerical instability encountered when calculatingthe wavefunction. In two dimensions an un-coupled matching states method and anasymptotic time dependent method are described.As an alternative to the above methods a coupled network theory is presented forthe first time which genuinely represents the 2D time independent electronicwavefunction. Nodes on the network are described by a unitary scattering matrixfrom which a 2D transfer matrix is derived, connecting lines on the network. Thescattering matrix for the whole system is created by combining the 2D scatteringmatrices for each line, themselves derived from the transfer matrices. The use of thescattering matrix is necessary to ensure numerical stability and currentconservation.It is shown that the bandstructure of the network is essential to creating agenuine 2D model whilst at the same time introducing a perturbing influence on themanifestation of physical phenomena. The advantages over other models is thecomplete absence of restriction on the potential profile considered and norequirement to separate the scalar energy and potential quantities into x and ycomponents. Also no problem with current continuity has been encountered. A majordisadvantage is the large time required to calculate wavefunctions compared with theun-coupled matching states method.The network is shown to reproduce the channel conductance quantization recentlyobserved experimentally and is in good agreement with both a 1D analytic model and a2D 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 1Dmodel. Also the wavefunction on resonance resembles one of the quasi states of thewell but with a phase shift.When applied to waveguides involving an interface between channels of differentwidths the network reveals a tendency for the wavefunction to relax to its originaltransverse state as it gets further from the interface. This tendency is mostpronounced for a tapered junction at low energy (energy of the order of the firsttransverse eigenvalue). The transmission coefficient for an abrupt junction displaysunusual dips above the quantization threshold of the narrow channel. Scattering intohigher modes is reduced both by reducing the ratio of channel widths and by reducingthe absolute lengths of the device.Finally circle and ring devices are studied, results displaying similarities withFinch's time dependent calculations. In particular scattering into the arms of the ringis observed to be mainly into the first mode if the energy is low and mainly into thethird mode if the energy is of the same order as the third transverse eigenvalue of thechannel. The tendency to relax into the original transverse state still operates overthe whole device.

[发布机构] University:University of Glasgow;Department:School of Engineering

[关键词] TK Electrical engineering. Electronics Nuclear engineering