  When an energy gap is induced in monolayer graphene the valley degeneracy is
broken and the energy spectrum of a confined system such as a quantum dot,
becomes rather complex exhibiting many irregular level crossings and small
energy spacings which are very sensitive to the applied magnetic field. Here we
study the energy spectrum of a graphene quantum dot that is formed between
Landau levels, and show that for the appropriate potential well the dot energy
spectrum in the first Landau gap can have a simple pattern with energies coming
from one of the two valleys only. This part of the spectrum has no crossings,
has specific angular momentum numbers, and the energy spacing can be large
enough, consequently, it can be probed with standard spectroscopic techniques.
The magnetic field dependence of the dot levels as well as the effect of the
mass-induced energy gap are examined, and some regimes leading to a
controllable quantum dot are specified. At high magnetic fields and negative
angular momentum a simple approximate method to the Dirac equation is developed
which gives further insight into the physics. The approximate energies exhibit
the correct trends and agree well with the exact energies.
