  Quantum oscillations can be used to determine properties of the Fermi surface
of metals by varying the magnitude and orientation of an external magnetic
field. Topological insulator surface states are an unusual mix of normal and
Dirac fermions. Unlike in graphene and simple metals, Berry's geometric phase
in topological insulator surface states is not necessarily quantised. We show
that reliably extracting this geometric phase from the phase offset associated
with the quantum oscillations is subtle. This is especially so in the presence
of a Dirac gap such as that associated with the Zeeman splitting or interlayer
tunneling. We develop a semi-classical theory for general mixed normal-Dirac
systems in the presence of a gap, and in doing so clarify the role of topology
and broken particle-hole symmetry. We propose a systematic procedure of fitting
Landau level index plots at large filling factors to reliably extract the phase
offset associated with Berry's phase.
