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Motivated by the recent experimental discovery of stronglysurface-plane-dependent superconductivity at surfaces of KTaO$_3$ singlecrystals, we calculate the electron-phonon coupling strength, $\lambda$, ofdoped KTaO$_3$ along the reciprocal-space high-symmetry directions. Using theWannier-function approach implemented in the EPW package, we calculate$\lambda$ across the experimentally covered doping range and compare itsmode-resolved distribution along the [001], [110] and [111] reciprocal-spacedirections. We find that the electron-phonon coupling is strongest in theoptical modes around the $\Gamma$ point, with some distribution to higher $k$values in the [001] direction. The electron-phonon coupling strength as afunction of doping has a dome-like shape in all three directions and itsintegrated total is largest in the [001] direction and smallest in the [111]direction, in contrast to the experimentally measured trends in criticaltemperatures. This disagreement points to a non-BCS character of thesuperconductivity. Instead, the strong localization of $\lambda$ in the softoptical modes around $\Gamma$ suggests an importance of ferroelectric soft-modefluctuations, which is supported by our findings that the mode-resolved$\lambda$ values are strongly enhanced in polar structures. The inclusion ofspin-orbit coupling has negligible influence on our calculated mode-resolved$\lambda$ values.

First-principles calculation of electron-phonon coupling in doped KTaO$_3$