  Low-dimensional electronic systems in thermoelectrics have the potential to
achieve high thermal-to-electric energy conversion efficiency. A key measure of
performance is the efficiency when the device is operated under maximum power
conditions. Here we study the efficiency at maximum power of three
low-dimensional, thermoelectric systems: a zero-dimensional quantum dot (QD)
with a Lorentzian transmission resonance of finite width, a one-dimensional
(1D) ballistic conductor, and a thermionic (TI) power generator formed by a
two-dimensional energy barrier. In all three systems, the efficiency at maximum
power is independent of temperature, and in each case a careful tuning of
relevant energies is required to achieve maximal performance. We find that
quantum dots perform relatively poorly under maximum power conditions, with
relatively low efficiency and small power throughput. Ideal one-dimensional
conductors offer the highest efficiency at maximum power (36% of the Carnot
efficiency). Whether 1D or TI systems achieve the larger maximum power output
depends on temperature and area filling factor. These results are also
discussed in the context of the traditional figure of merit $ZT$.
