  Optical and electronic phenomena in solids arise from the behaviour of
electrons and holes (unoccupied states in a filled electron sea). Electron-hole
symmetry can often be invoked as a simplifying description, which states that
electrons with energy above the Fermi sea behave the same as holes below the
Fermi energy. In semiconductors, however, electron-hole symmetry is generally
absent since the energy band structure of the conduction band differs from the
valence band. Here we report on measurements of the discrete, quantized-energy
spectrum of electrons and holes in a semiconducting carbon nanotube. Through a
gate, an individual nanotube is filled controllably with a precise number of
either electrons or holes, starting from one. The discrete excitation spectrum
for a nanotube with N holes is strikingly similar to the corresponding spectrum
for N electrons. This observation of near perfect electron-hole symmetry
demonstrates for the first time that a semiconducting nanotube can be free of
charged impurities, even in the limit of few-electrons or holes. We furthermore
find an anomalously small Zeeman spin splitting and an excitation spectrum
indicating strong electron-electron interactions.
