Exactly soluble two-dimensional electron gas in a magnetic-field barrier

The single-particle energy eigenstates of a two-dimensional electron gas confined to the x-y plane and in the presence of an external-magnetic-field barrier whose functional form is B(x,y) =${\mathit{B}}_0$(1-${\tanh }^2$x/d)z^, with ${\mathit{B}}_0$ and d arbitrary, is solved exactly. It is found that the spectrum has bounded and unbounded states. The former are confined to the region where the magnetic field is appreciable. The lowest-lying eigenstates resemble the Landau levels of the constant-field case, but they also drift along the y axis with a speed proportional to the magnetic-field gradient. The unbounded states are extended either on one side of the barrier or on both sides, depending on their energy and asymptotic momenta. It is found that the discrete and continuum spectra overlap in an energy range. It is also argued that these results apply qualitatively to a general class of magnetic-field barriers.