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We describe the effect of a magnetic field on the current-voltage characteristics of semiconductor tunnel barriers in which superlattices are used to filter the initial and final states. A field applied parallel to the layers was found to destroy the negative differential conductance intrinsic to such structures, with a zerobias anomaly appearing in high fields. A qualitative explanation of this, which considers the effect of the field on the electron states in the superlattice and on the tunnelling process, is presented.