  Advances in solid-state device design now allow the spectrum of transmitted
electrons in thermionic and thermoelectric devices to be engineered in ways
that were not previously possible. Here we show that the shape of the electron
energy spectrum in these devices has a significant impact on their performance.
We distinguish between traditional thermionic devices where electron momentum
is filtered in the direction of transport only and a second type, in which the
electron filtering occurs according to total electron momentum. Such 'total
momentum filtered' kr thermionic devices could potentially be implemented in,
for example, quantum dot superlattices. It is shown that whilst total momentum
filtered thermionic devices may achieve efficiency equal to the Carnot value,
traditional thermionic devices are limited to efficiency below this. Our second
main result is that the electronic efficiency of a device is not only improved
by reducing the width of the transmission filter as has previously been shown,
but also strongly depends on whether the transmission probability rises sharply
from zero to full transmission. The benefit of increasing efficiency through a
sharply rising transmission probability is that it can be achieved without
sacrificing device power, in contrast to the use of a narrow transmission
filter which can greatly reduce power. We show that devices which have a
sharply-rising transmission probability significantly outperform those which do
not and it is shown such transmission probabilities may be achieved with
practical single and multibarrier devices. Finally, we comment on the
implications of the effect the shape of the electron energy spectrum on the
efficiency of thermoelectric devices.
