In condensed matter physics, first-principles methodologies play a significant role in understanding various interesting phenomena and physical laws, such as electronic and optoelectronic properties of crystals. And low dimensional materials exhibit many extraordinary physical properties due to quantum confinement, enhanced interaction, and special screening effects, and so on. Quasi-one-dimensional graphene nanoribbons (GNRs) are a promising new platform for future nanoelectronics applications, and the recent booming development of bottom-up synthesis techniques enables atomically precise synthesis of GNRs, so that GNRs of different widths, edge shapes and dopants could be well-studied.

This dissertation mainly focuses on the theoretical and computational study of the topological electronic properties of GNRs, the substrate interaction of doped GNRs, the design of a double-quantum-well GNR transistor, and the optical properties of conjugated polymers. This dissertation is organized as follows:

• In Chapter 1, we review the theoretical and computational methodologies that we used to calculate the electronic and optical properties of relevant low-dimensional materials in the dissertation, including density functional theory (DFT) for ground-state properties as well as the GW and the GW plus Bethe-Salpeter equation (GW-BSE) methods for excited-state properties.

• In Chapter 2, we present our study on the topological phases of armchair, cove-edged, and chevron GNRs, as well as GNRs with dopants by first-principles calculations and model-related methods. We further present a novel design of a periodically doped GNR structures with electric ﬁeld tunable topological properties, which may have potential application in nanoelectronics.

• In Chapter 3, we show our detailed investigation of the bulk-edge correspondence in Z2 classification of quasi 1D systems by using model and first-principles calculations.

• In Chapter 4, we present a first-principles study of electronic properties of GNRs collaborated with the experimental groups. We investigate the strong interaction between the metallic substrate and the dopant states of the boron-doped armchair GNR for different doping concentrations. Strong substrate interactions have been observed both by first-principles calculations and scanning tunneling spectroscopy (STS) measurements.

• Chapter 5 focuses on an application study of the GNRs. We design a double-quantum-well GNR transistor and analyze the possibility of using such a device to break the thermal limit in the swing voltage in order to build energy efficient transistors. We test the validity of a double-quantum-well GNR transistor, and obtain the analytical form of spectral lineshapes of a quantum dot state coupled to a metallic wire with ﬁnite band width. The findings shown in this chapter reveal the theoretical principles in designing nanoelectronic transistors with low swing voltages.

• In Chapter 6, we present our first-principles study on the optical and excitonic properties of conjugated polymers using the GW-BSE methodology. We present detailed investigations on different approximations used in density functional theory and GW-BSE calculations and their effects to the calculated results. The new calculated absorption spectrum with improved methodologies agrees much better with the experiments than previous reported first-principles results. We also present a new development on the interface code bridging the BerkeleyGW code with an ab initio DFT code to enable GW-BSE calculations starting from DFT using metaGGA and hybrid exchange-correlation functionals.