  We report thermal conductivity measurements on single crystals of
herbertsmithite, over a wide range of temperatures (0.05-120 K) in magnetic
fields up to 15 T. We also report measurements of the thermal Hall effect,
found to be vanishingly small. At high temperatures, in the paramagnetic
regime, the thermal conductivity has a negligible field dependence. Upon
cooling and the development of correlations, the onset of a clear monotonic
field dependence below about 20 K signals a new characteristic temperature
scale that may reflect the subtle crossover towards the quantum spin liquid
regime. Deconfined spinons, if present, are not detected and phonons, as the
main carriers of heat, are strongly scattered by the intrinsic spin excitations
and the magnetic defects. In view of the colossal fields required to affect the
intrinsic spins, most of the field-induced evolution is attributed to the
progressive polarization of some magnetic defects. By elaborating a
phenomenological model, we extract the magnetization of these main scattering
centers which does not resemble the Brillouin function for free spins 1/2,
requiring to go beyond the paradigm of isolated paramagnetic spins. Besides,
the onset of a nonmonotonic field dependence below about 2 K underlines the
existence of another characteristic temperature scale, previously highlighted
with other measurements, and sheds new light on the phase diagram of
herbertsmithite down to the lowest temperatures.
