Carbon nanotubes provide a unique system for studying one-dimensional quantization phenomena. Scanning tunneling microscopy was used to observe the electronic wave functions that correspond to quantized energy levels in short metallic carbon nanotubes. Discrete electron waves were apparent from periodic oscillations in the differential conductance as a function of the position along the tube axis, with a period that differed from that of the atomic lattice. Wave functions could be observed for several electron states at adjacent discrete energies. The measured wavelengths are in good agreement with the calculated Fermi wavelength for armchair nanotubes.

Scanning tunneling microscope spectroscopy is used to study in detail the electronic band structure of carbon nanotubes as well as to locally investigate electronic features of interesting topological sites such as nanotube ends and bends. From a large number of measurements of the tunneling density-of-states ͑DOS͒ nanotubes can be classified, according to predictions, as either semiconducting ͑two-third of the total number of tubes͒ or metallic ͑one-third͒. The energy subband separations in the tunneling DOS compare reasonably well to theoretical calculations. At nanotube ends, spatially resolved spectra show additional sharp conductance peaks that shift in energy as a function of position. Spectroscopy measurements on a nanotube kink suggest that the kink is a heterojunction between a semiconducting and a metallic nanotube.

We present a technique to control the length of carbon nanotubes. Individual carbon nanotubes can be locally cut by applying a voltage pulse to the tip of a scanning tunneling microscope (STM). Topographic imaging and STM spectroscopy are subsequently used to investigate the result. The electronic properties of a nanotube can be strongly changed by reducing the size. Current-voltage curves obtained by STM spectroscopy on a 30 nm short tube created from a longer nanotube show a stepwise increase of the current, which is attributed to quantum size effects.

Low-temperature scanning tunneling spectroscopy measurements on clean InAs͑110͒ surfaces demonstrate the possibility to investigate Landau quantization with subnanometer resolution. Separate Landau levels are resolved at magnetic fields of 2 T and larger. Experiments with different tips show a significant tip dependence.In zero field resonances are observed near the onset of the conduction band, which are attributed to tip-induced band bending. Although Landau quantization is only present parallel to the sample surface, the Landau levels give a large contribution to the total tunnel current.