From a theoretical point of view, Hartke's expression has to be preferred to Poole–Frenkel equation on the grounds that the threedimensionality of the problem of the trap barrier lowering is considered. Additional three-dimensional models have been developed, differing by the treatment they do of the emission process in the upwind direction. Ieda, Sawa, and Kato, for example, have proposed a model that includes the barrier variation in directions both forward and opposite to the electric field.