  SrTiO$_3$, a quantum paralectric, displays a detectable phonon thermal Hall
effect (THE). Here we show that the amplitude of THE is extremely sensitive to
stoichiometry. It drastically decreases upon substitution of a tiny fraction of
Sr atoms with Ca, which stabilizes the ferroelectric order. It drastically
increases by an even lower density of oxygen vacancies, which turn the system
to a dilute metal. The enhancement in the metallic state exceeds by far the sum
of the electronic and the phononic contributions. We explain this observation
as an outcome of three features: i) heat is mostly transported by phonons; ii)
the electronic Hall angle is extremely large; and iii) there is substantial
momentum exchange between electrons and phonons. Starting from Herring's
picture of phonon drag, we arrive to a quantitative account of the enhanced
THE. Thus, phonon drag, hitherto detected as an amplifier of thermoelectric
coefficients, can generate a purely thermal transverse response in a dilute
metal with a large Hall angle. Our results reveal a hitherto unknown
consequence of momentum-conserving collisions between electrons and phonons.ely
thermal transverse response in a dilute metal with a large Hall angle. Our
results reveal a hitherto unknown consequence of momentum-conserving collisions
between electrons and phonons.
