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Some semiconductor devices such as lasers have long had critical dimensions on the nanoscale where quantum effects are critical. Others such as MOSFETs are now being scaled to within this regime. Quantum effects neglected in semiclassical models become increasing important at the nanoscale. Meanwhile, scattering remains important even in MOSFETs of 10 nm and below. Therefore, accurate quantum transport simulators with scattering are needed to explore the essential device physics at the nanoscale. The work of this dissertation is aimed at developing accurate quantum transport simulation tools for deep submicron device modeling, as well as utilizing these simulation tools to study the quantum transport and scattering effects in the nano-scale semiconductor devices. The basic quantum transport method "Schrödinger Equation Monte Carlo" (SEMC) provides a physically rigorous treatment of quantum transport and phasebreaking inelastic scattering (in 3D) via real (actual) scattering processes such as optical and acoustic phonon scattering. The SEMC method has been used previously to simulate carrier transport in nano-scaled devices in order to gauge the potential reliability of semiclassical models, phase-coherent quantum transport, and other limiting models as the transition from classical to quantum transport is approached. In this work, SEMC-1D and SEMC-2D versions with long range polar optical scattering processes have been developed and used to simulate quantum transport in tunnel injection lasers and nanoscaled III-V MOSFETs. Simulation results serve not only to demonstrate the capabilities of the developed quantum transport simulators, but also to illuminate the importance of physically accurate simulation of scattering for the predictive modeling of transport in nano-scaled devices.

This dissertation considers the effect of measuring randomly varying local hydraulic conductivities K(x) on one's ability to predict steady state flow within a bounded domain, driven by random source and boundary functions. That is, the work concerns the prediction of local hydraulic head h(x) and Darcy flux q(x) by means of their unbiased ensemble moments (h(x))(κ) and (q(x))(κ) conditioned on measurements of K(x). These predictors satisfy a deterministic flow equation in which (q(x))(κ) = -(κ)(x)∇(h(x))(κ) + r(κ)(x) where κ(x) is a relatively smooth unbiased estimate of K(x) and r(κ)(x) is a "residual flux." A compact integral expression is derived for r(κ)(x) which is rigorously valid for a broad class of K(x) fields, including fractals. It demonstrates that (q(x))(κ) is nonlocal and non-Darcian so that an effective hydraulic conductivity does not generally exist. It is shown analytically that under uniform mean flow the effective conductivity may be a scalar, a symmetric or a nonsymmetric tensor, or a set of directional scalars which do not form a tensor. For cases where r(κ)(x) can neither be expressed nor approximated by a local expression, a weak (integral) approximation (closure) is proposed, which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of K(x) measurements increase. The nonlocal deterministic flow equation can be solved numerically by standard methods; the theory here shows clearly how the scale of grid discretization should relate to the scale, quantity and quality of available data. After providing explicit approximations for the prediction error moments of head and flux, some practical methods are discussed to compute κ(x) from noisy measurements of K(x) and to calculate required second moments of the associated estimation errors when K(x) is log normal. Nonuniform mean flow is studied by conducting high resolution Monte Carlo simulations of two dimensional seepage to a point sink in statistically homogeneous and isotropic log normal K(x) fields. These reveal the existence of radial effective hydraulic conductivity which increases from the harmonic mean of K(x) near interior and boundary sources to geometric mean far from such sources for σ^2/Υ (the variance of ln K) at least as large as 4. They suggest the possibility of replacing r(κ)(x) by a local expression at distances of few conditional integral scales from the interior and boundary sources. Special attention is paid to the "art" of random field generation, and comparisons are made between four alternative methods with five different random number generators.

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Monte Carlo modelling is a common technique in numerous fields, and is widely used in semiconductor device simulation. This thesis describes the application of Monte Carlo modelling to the simulation of Gunn diode devices, focusing on devices composed of Gallium Arsenide (GaAs) and Gallium Nitride (GaN). Gunn diodes are simple structures that take advantage of negative differential resistance to act as a source of high frequency radiation, from 10 GHz to over 100 GHz in GaAs devices. It has been theorised that GaN should exhibit negative differential resistance and a GaN Gunn diode could produce radiation of even higher frequency, within the terahertz band. Gunn diodes have the advantage of being cheap and portable, and so are worth exploring as such a source. Unfortunately, GaN devices have a high electron density and so they tend to generate heat quickly. It therefore becomes important to include modelling of heat generation and flow in simulations of these devices. This is uncommon in Monte Carlo models of Gunn diodes, as in less highly doped devices thermal effects can usually be assumed to result in the device reaching an equilibrium temperature of about 100 K above the ambient. This thesis describes the creation of a model to track the generation and distribution of heat during operation of a GaN device. Simulations found that thermal effects within the device were significant. Heat generation occurred to the extent that the device could only be operated in pulsed mode, with on pulses of 2 ns requiring 50 ns of cooling for sustainable operation. The increased temperature within the device also lead to deleterious changes in the Gunn diode's operating frequency. In the simulated device, a 150 K change in temperature lead to a decrease in operating frequency of 40 GHz, from an initial frequency of 280 GHz. At the end of 2 ns of operation, the mean temperature within the device had increased by 120 K. The high accidental doping level in GaN also means the use of a doping notch to act as a nucleation point for dipoles within a Gunn diode, a common technique in other materials, becomes less feasible. As an alternative to a notch, a device was simulated incorporating a doping spike to nucleate the dipole. The use of a doping spike is not novel, however its use in GaN has not been previously explored. Simulations found that a fully-depleted p-type doping notch of length 2.1 nm, doped at 1x1024 m-3 would act as a nucleation point for dipole operation. The device was compared to a simulated device incorporating a doping notch of width 0.25 µm doped at 0.5x1023 m-3 and found to operate at a similar frequency and RF efficiency, making it a viable substitute. One limitation of Gunn diodes is that when operated in transit-time mode, the operating frequency is determined by the length of the diode's transit region and so is well-defined and fixed. This means that traditional Gunn diodes are not as useful a source of radiation for spectroscopic applications as might be desirable. Recent experimental results for planar devices have shown a broadening in operation frequency and even multiple frequencies. This thesis explores the hypothesis that such a broadening might be achieved in a vertical structure via the incorporation of an additional notch into the Gunn diode's transit region, effectively incorporating two transit regions into the device. Results showed that this novel device structure did show multiple modes of operation. Under a DC applied voltage, the device showed spontaneous switching behaviour, oscillating between dipole and accumulation layer operation from the second notch. Changes in the frequency of an applied RF voltage would shift the device from operating from the first or second notch, in dipole and accumulation layer mode respectively.

published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy

Decreasing block pricing refers to the practice of selling a product at successively lower marginal prices as the amount purchased in any one time period increases. In more familiar terms, this practice can be thought of as any quantity discount scheme as long as marginal price does not vary continuously with quantity. Decreasing block pricing results in a faceted, non-convex budget set, and under standard assumptions concerning consumer preferences, yields several nonstandard theoretical implications. The central goal of this paper is to formulate an estimation technique which is consistent with these implications. When the budget set is not convex, the uniqueness of consumer equilibrium is no longer guaranteed. It also follows that discontinuities in demand occur whenever consumer equilibrium shifts from one facet of the budget constraint to another. Prior empirical studies have not made use of demand functions consistent with these results. In Chapter 2, a utility-maximizing algorithm was developed to determine consumer equilibrium given the declining block pricing schedule and income for a Cobb-Douglas utility function. In developing this algorithm, it was made clear that the proper approach for estimating individual demand was through the use of a block-dependent independent variable. The coefficient of this block-department independent variable provided an estimate of a utility function parameter which completely specified the Cobb-Douglas form. Incorporating this utility function estimate into the utility-maximation algorithm made it possible to obtain estimates of consumption given changes in any or all of the rate schedule components. While the use of a block-dependent independent variable is the theoretically correct method for estimating demand, it poses an inescapable problem of errors-in-variables. A Monte Carlo study was performed in Chapter 2 to investigate, among other things, the seriousness of the errors-in-variables bias. The results were quite encouraging. When using data incorporating extremely large error variances, amazingly precise estimates were obtained. Another encouraging Monte Carlo result was when comparing samples not containing a discontinuity with those with one, it was found that the latter produced estimates with statistically significant superiority. Chapter 3 generalized the estimation technique of the previous chapter to allow the estimation of demand using cross-sectional data. The data base recorded monthly electricity consumption for households from a number of cities whose utilities had decreasing block rates. Seven of these cities were selected for analysis. The data also included various demographic characteristics and electric appliance stock information. The generalization was accomplished by assuming that all households had a Stone-Geary utility function. Also, the utility function parameter representing the minimum required quantity of electricity was assumed to depend linearly on the household's appliance stock and demographic characteristics. This allowed demand to vary across households on the basis of this parameter and income. The results of applying this regression technique to the cross-sectional data were then compared with results from a conventional, non-theoretically based demand specification. The data were used in pooled and individual month form with the former yielding much better statistical results. The Stone-Geary form provided a greater number of significant coefficients for price and income variables than the conventional version. The predominant failure of the conventional version was that the coefficient of marginal price was rarely significant and when significant, frequently of the wrong sign. For the same samples, the Stone-Geary results were quite acceptable except for the regressions involving one of the cities. Thus, it was demonstrated that a method consistent with the theoretical implications of decreasing block pricing is easily applied to cross-sectional data and produces better results than conventional techniques.

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