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It has recently become clear that phonons generate a sizable thermal Halleffect in cuprates, whether they are undoped, electron-doped or hole-doped(inside the pseudogap phase). At higher doping, where cuprates are reasonablygood metals, mobile electrons also generate a thermal Hall effect, the thermalequivalent of the standard electrical Hall effect. Here we show that in thecleanest crystals of the electron-doped cuprate Nd$_{2-x}$Ce$_{x}$CuO$_{4}$, athigh doping, the phonon and electron contributions to the thermal Hallconductivity $\kappa_{\rm {xy}}$ are of comparable magnitude, but of oppositesign. In samples of lower quality, phonons dominate $\kappa_{\rm {xy}}$,resulting in a negative $\kappa_{\rm {xy}}$ at all temperatures. The fact thatthe negative phononic $\kappa_{\rm {xy}}$ in the metallic state is similar inmagnitude and temperature dependence to that found in the insulating state atlower doping rules out any mechanism based on skew scattering of phonons offcharged impurities, since a local charge should be screened in the metallicregime. The phononic $\kappa_{\rm {xy}}$ is found to persist over the entiredoping range where antiferromagnetic correlations are known to be significant,suggesting that such correlations may play a role in generating the phononthermal Hall effect in electron-doped cuprates. If the same mechanism is alsoat play in hole-doped cuprates, the presence of a phononic $\kappa_{\rm {xy}}$below (and only below) the critical doping $p^{\star}$ would be evidence thatspin correlations are a property of the pseudogap phase.

Thermal Hall conductivity of electron-doped cuprates: Electrons and phonons