  The question of whether relaxing momentum conservation can increase the
performance of thermionic cooling device is examined. Both homojunctions and
heterojunctions are considered. It is shown that for many cases, a
non-conserved lateral momentum model overestimates the current. For the case of
heterojunctions with a much heavier effective mass in the barrier and with a
low barrier height, however, non-conservation of lateral momentum may increase
the current. These results may be simply understood from the general principle
that the current is limited by the location, well or barrier, with the smallest
number of conducting channels. These results also show that within thermionic
emission framework, the possibilities of increasing thermionic cooling by
relaxing momentum conservation are limited. More generally, however, when the
connection to the source is weak or in the presence of scattering, the
situation may be different. Issues that deserve further study are identified.
