  Measurements of the thermal Hall conductivity in hole-doped cuprates have
shown that phonons acquire chirality in a magnetic field, both in the pseudogap
phase and in the Mott insulator state. The microscopic mechanism at play is
still unclear. A number of theoretical proposals are being considered,
including skew scattering of phonons by various defects, the coupling of
phonons to spins, and a state of loop-current order with the appropriate
symmetries, but more experimental information is required to constrain
theoretical scenarios. Here we present our study of the thermal Hall
conductivity $\kappa_{\rm {xy}}$ in the electron-doped cuprates
Nd$_{2-x}$Ce$_x$CuO$_4$ and Pr$_{2-x}$Ce$_x$CuO$_4$, for dopings across the
phase diagram, from $x$ = 0, in the insulating antiferromagnetic phase, up to
$x$ = 0.17, in the metallic phase above optimal doping. We observe a large
negative thermal Hall conductivity at all dopings, in both materials. Since
heat conduction perpendicular to the CuO$_2$ planes is dominated by phonons,
the large thermal Hall conductivity we observe in electron-doped cuprates for a
heat current in that direction must also be due to phonons, as in hole-doped
cuprates. Measurements with a heat current perpendicular to the CuO$_2$ planes
confirm that phonons are responsible for this thermal Hall signal, as in
hole-doped cuprates. However, the degree of chirality, measured as the ratio
$|\kappa_{\rm {xy}}$ / $\kappa_{\rm {xx}} |$, where $\kappa_{\rm {xx}}$ is the
longitudinal thermal conductivity, is much larger in the electron-doped
cuprates. We discuss various factors that may be involved in the mechanism that
confers chirality to phonons in cuprates, including short-range spin
correlations.
