Fluctuations Due to Object Discreteness

Citation

Boudville, Wesley J. (1988) Fluctuations Due to Object Discreteness. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/drt3-m876. https://resolver.caltech.edu/CaltechTHESIS:01182013-142410145

Abstract

In this thesis I discuss the effect of randomness or fluctuations in topics drawn from three different areas of condensed matter physics: Metal-semiconductor ohmic contacts, the planar-doped barrier transistor and two dimensional continuum percolation.

Chapter 1 contains an introduction to the thesis. It outlines and summarises the rest of the thesis.

Chapters 2 and 3 describe the work done on metal-semiconductor ohmic contacts. The motivation was to find a possible limit to the fabrication of Very Large Scale Integrated circuits (VLSI). These chips are powered by current entering the chip along metal lines. A metal line ultimately makes ohmic contact to a semiconductor. This is done by doping the semiconductor heavily. In general, it is desirable for the contacts to have as low a resistance as possible, in order to reduce voltage drops across the contact. Furthermore, the conductances per unit area of such contacts should be the same, otherwise quite different currents could flow through two contacts of nominally the same specific conductance. In modelling such a contact, it has previously been assumed that in the depletion region of the semiconductor, at the junction, the dopants form an ionised continuum. However, dopants are discrete. Hence, over the small distance (~ 100 Å) of the depletion region of a heavily doped semiconductor, the current will see a spatially randomly varying potential. This causes the resistance of a contact to vary due to the random configuration of dopants in the contact.

In Chapter 2, a continuum model of the junction is presented. This is considered an improvement over that used in the literature for ohmic contacts. Chapter 3 describes how the discreteness of the dopants and their random distribution is taken into account. Simulations are made to find the resistance fluctuations of a contact, given the size of the contact and the doping in the semiconductor. I investigate how the fluctuations scale as a function of the contact size and doping. Also considered is the effect of making the current trajectories three dimensional, instead of restricting them to being normal to the junction. It is found that there is little modification to the one dimensional nature of the trajectories. (For the purposes of obtaining actual numbers, the semiconductor was chosen to be n-type GaAs.) Simulations indicate that the dopant discreteness will not be a problem in ohmic contacts in VLSI. Rather, it will be a problem for Ultra Large Scale Integrated circuits (ULSI). The resistance fluctuations become significant for contact sizes on the order of 1000 Å. Currently, the semiconductor industry is just at the submicron level. It will probably be at least the mid 1990s before the device sizes approach 1000 Å.

Chapter 4 contains suggestions of future research based on the results of Chapters 2 and 3.

In Chapter 5 an analytic means of estimating the probability distribution of barrier heights in the barrier regions of a planar doped barrier transistor is derived. The barrier heights vary spatially due to the random distribution of discrete dopants in the barrier regions, much as in the metal-semiconductor contact of the earlier chapters. The resultant distribution is compared to one found from a three dimensional finite element simulation in the literature. The agreement is good. The analytic results presented go beyond this by showing the dependence of the barrier fluctuations on the doping and the thickness of the barrier regions.

Chapter 6 describes finite size effects in two dimensional anisotropic continuum percolation. Continuum percolation is where the objects that percolate are placed randomly in a given region, as distinct from percolation on a lattice. The anisotropy refers to the objects having a preferred average orientation. In two (and three) dimensions it has been found that the critical lengths for the onset of percolation are different for percolation along the average object orientation or transverse to this. There is a universal behaviour to this dependence of critical lengths on the number of objects in the sample and on their degree of orientation. I have developed a theory to explain this. It agrees well with simulation results.

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