  Spatially-resolved heating and cooling in nanostructures is nowadays measured
with various nanoscale thermometry techniques, including scanning thermometry.
Yet the most commonly used theory of nanoscale heating and thermoelectricity --
Landauer scattering theory -- is not appropriate to model such measurements.
Hence, we analyze a minimal model of spatially-resolved heat transfer between
electrons and phonons in simple thermoelectric nanostructures. This combines
Landauer scattering formalism with a Boltzmann equation for transport,
revealing the non-locality of Joule heating and Peltier cooling induced by a
scatterer in a nanowire. The corresponding heating or cooling of the phonons is
caused by the voltage drop at the scatterer, but is often maximal at a certain
distance from the scatterer. This distance is of the order of the
electron-phonon scattering length. Scanning thermal microscopy, such as
SQUID-on-tip thermometers, should detect this non-locality as phonon hot spots
and cold spots, spatially separated from the scatterer. We provide physical
arguments explaining the thermoelectric response of the combined system of wire
and scatterer, and in particular, why the resulting heating and cooling is
sometimes the opposite to that predicted by the standard Landauer scattering
theory.
