Electronics have become a fundamental aspect of modern society and smaller, even more efficient, transistors form the backbone of the electronics industry of the future. Laterally Diffused Metal-Oxide-Semiconductor (LDMOS) field-effect transistors are a class of transis- tors commonly used in every electronic device we use in our daily life such as smart-phone chargers, kitchen appliances, and autonomous vehicles. In this thesis, we present a systematic investigation of device design optimization for LDMOS transistors suitable for low-voltage (< 30 V) and mid-voltage (30 V – 100 V) power applications. We perform a numerical study as well as an analytical investigation to find the theoretical limits of LDMOS tran- sistors. We target critical transistor characteristics such as subthreshold leakage current, breakdown voltage, on-resistance, and figure-of-merit to perform a fundamental analysis. For low-voltage applications, we perform a channel length study to find the optimum chan- nel length, resulting in the minimum on-resistance, to achieve the minimum device size with maximum efficiency. We use a Technology Computer-Aided Design (TCAD) commercial drift-diffusion simulation package to perform our numerical study. Moreover, we develop an in-house computer code to automate the process of device design and optimization. For mid-voltage applications, we analytically and numerically investigate the performance of LDMOS transistors with different field-oxide configurations. We derive a new analytical relation between breakdown voltage and on-resistance quantifying the fundamental limits of the trade-off between resistance and current associated with the drift region in an LDMOS with field oxides. We find the optimized device characteristics, such as drift doping concen- tration, which minimizes the on-resistance. We finally verify our analytical findings with a large number of numerical simulations modeled in TCAD. We also present a first-principles investigation to discover novel two-dimensional dielectric materials used in two-dimensional transistors. We calculate the critical properties of dielectric materials such as in-plane and out-of-plane dielectric constants, bandgap energy, exfoliation energy, and equivalent oxide thickness.