This dissertation provides a brief introduction into scanning tunneling microscopy, and then Chapter III reports on the design and operation of a cryogenic ultra-high vacuum scanning tunneling microscope (STM) coupled to a closed-cycle cryostat (CCC). The STM is thermally linked to the CCC through helium exchange gas confined inside a volume enclosed by highly flexible rubber bellows. The STM is thus mechanically decoupled from the CCC, which results in a significant reduction of the mechanical noise transferred from the CCC to the STM. Noise analysis of the tunneling current shows current fluctuations up to 4% of the total current, which translates into tip-sample distance variations of up to 1.5 picometers. This noise level is sufficiently low for atomic-resolution imaging of a wide variety of surfaces. To demonstrate this, atomic-resolution images of Au(111) and NaCl(100)/Au(111) surfaces, as well as of carbon nanotubes deposited on Au(111), were obtained. Other performance characteristics such as thermal drift analysis and a cool-down analysis are reported. Scanning tunneling spectroscopy (STS) measurements based on the lock-in technique were also carried out and showed no detectable presence of noise from the CCC. These results demonstrate that the constructed CCC-coupled STM is a highly stable instrument capable of highly detailed spectroscopic investigations of materials and surfaces at the atomic-scale. A study of electron transport in single-walled carbon nanotubes (SWCNTs) was also conducted. In Chapter IV, STS is used to study the quantum-confined electronic states in SWCNTs deposited on the Au(111) surface. The STS spectra show the vibrational overtones which suggest rippling distortion and dimerization of carbon atoms on the SWCNT surface. This study experimentally connects the properties of well-defined localized electronic states to the properties of their associated vibronic states. In Chapter V, a study of PbS nanocrystals was conducted to study the effect of localized sub-bandgap states associated with surface imperfections. A correlation between their properties and the atomic-scale structure of chemical imperfections responsible for their appearance was established to understand the nature of such surface states. This dissertation includes both previously published/unpublished and co-authored material.

Cold, trapped atomic ions have enabled the investigation of fundamental physics and generated a rich field of applications. Foremost among these is quantum computation which has recently driven the development of the sophisticated, scalable surface-electrode trap. Despite the many advantages of surface-electrode traps, the typically smaller ion-electrode distance, d, in these traps results in an increased ion heating rate that is proportional to d^(-4) and a decreased trap well-depth that is proportional to d^(-2). These shortcomings can be simultaneously addressed by installing the trap into a cryogenic environment. With this in mind, a closed-cycle, cryogenic ion trapping apparatus that maintains excellent vacuum, is highly modular, has increased optical access, and uses a simple vibration isolation system has been developed. Single ions are trapped and used to characterize system properties such as the motion of the vibration isolation stage. In order to compare this system to a similar room temperature apparatus, the ion trapping lifetime and heating rate are determined. A single ion also serves as a sensitive electric field probe that is used to measure and compensate stray electric fields across the trap. Due to the long dark ion lifetimes in this system, it is well-suited to probing the stability of small, linear ion crystals. Linear ion crystals of arbitrary length are built in an automated fashion using transport waveforms and the scaling of dark lifetime with ion number for N <= 6 is investigated. These data are then used to consider the relevance of various loss channels.