Fermions in a graphene sheet behave like massless particles. We show that by folding the sheet into a tube they acquire non-zero effective mass as they move along the tube axis. That is, changing the space topology of graphene from 2D to 1D (space compactification) changes the 2D massless problem into an effective massive 1D problem. The size of the resulting mass spectrum depends on the quantized azimuthal frequency and its line spacing is proportional to the inverse of the tube diameter.
In the past, the study of relativistic particles has been the exclusive domain of
high-energy and particle physics. In graphene, nonetheless, the linear electronic band
dispersion near the Dirac points gave rise to charge carriers (electrons or holes) that
propagate as if they were massless fermions with speeds of the order of 106 m/s rather
than the speed of light 3108 m/s. Hence, charge carriers in this structure should be
described by the massless Dirac equation rather than the usual Schrodinger equation.
The physics of relativistic electrons is thus now experimentally accessible in graphenebased
solid-state devices, whose behavior differs drastically from that of similar devices
fabricated with usual semiconductors. Consequently, new unexpected phenomena have
been observed while other phenomena that were well-understood in common semiconductors,
such as the quantum Hall effect and weak-localization, exhibited surprising
behavior in graphene. Thus, graphene devices enabled the study of relativistic dynamics
in controllable nano-electronic circuits (relativistic electrons on-a-chip) and their
behavior probes our most basic understanding of electronic processes in solids. It also
allowed for the observation of some subtle effects, previously accessible only to high
energy physics, such as Klein tunneling and vacuum breakdown.