  We present exact analytical solutions for the zero-energy modes of
two-dimensional massless Dirac fermions fully confined within a smooth
one-dimensional potential V(x)= - {\alpha}/cosh({\beta}x), which provides a
good fit for potential profiles of existing top-gated graphene structures. We
show that there is a threshold value of the characteristic potential strength
{\alpha}/{\beta} for which the first mode appears, in striking contrast to the
non-relativistic case. A simple relationship between the characteristic
strength and the number of modes within the potential is found. An experimental
setup is proposed for the observation of these modes. The proposed geometry
could be utilized in future graphene-based devices with high on/off current
ratios.
