  It is well known that the electronic thermal conductivity of clean
compensated semimetals can be greatly enhanced over the electric conductivity
by the availability of an ambipolar mechanism of conduction, whereby electrons
and holes flow in the same direction experiencing negligible Coulomb scattering
as well as negligible impurity scattering. This enhancement -- resulting in a
breakdown of the Wiedemann-Franz law with an anomalously large Lorenz ratio --
has been recently observed in two-dimensional monolayer and bilayer graphene
near the charge neutrality point. In contrast to this, three-dimensional
compensated semimetals such as WP$_2$ and Sb are typically found to show a
reduced Lorenz ratio. This dramatic difference in behavior is generally
attributed to different regimes of Fermi statistics in the two cases:
degenerate electron-hole liquid in compensated semimetals versus non-degenerate
electron-hole liquid in graphene. We show that this difference is not
sufficient to explain the reduction of the Lorenz ratio in compensated
semimetals. We argue that the solution of the puzzle lies in the ability of
compensated semimetals to sustain sizeable regions of electron-hole
accumulation near the contacts, which in turn is a consequence of the large
separation of electron and hole pockets in momentum space. These accumulations
suppress the ambipolar conduction mechanism and effectively split the system
into two independent electron and hole conductors. We present a quantitative
theory of the crossover from ambipolar to unipolar conduction as a function of
the size of the electron-hole accumulation regions, and show that it naturally
leads to a sample-size-dependent thermal conductivity.

Mesoscale and Nanoscale Physics