Search for ${D}^{0}\to {K}^{-}\eta {e}^{+}{\nu }_{e}$, ${D}^{+}\to {K}_{S}^{0}\eta {e}^{+}{\nu }_{e}$ and ${D}^{+}\to \eta \eta {e}^{+}{\nu }_{e}$ decays

By analyzing ${e}^{+}{e}^{-}$ annihilation data corresponding to an integrated luminosity of $7.93\text{}\text{}{\mathrm{fb}}^{-1}$, collected at the center-of-mass energy of 3.773 GeV with the BESIII detector, we search for the semileptonic decays ${D}^{0}\to {K}^{-}\eta {e}^{+}{\nu }_{e}$, ${D}^{+}\to {K}_{S}^{0}\eta {e}^{+}{\nu }_{e}$ and ${D}^{+}\to \eta \eta {e}^{+}{\nu }_{e}$ for the first time. We present evidence for ${D}^{0}\to {K}^{-}\eta {e}^{+}{\nu }_{e}$ with a significance of $3.3\sigma $. The branching fraction of ${D}^{0}\to {K}^{-}\eta {e}^{+}{\nu }_{e}$ is measured to be $\left(0.8{4}_{-0.34}^{+0.29}±0.22\right)×{10}^{-4}$. No significant signals are observed for the decays ${D}^{+}\to {K}_{S}^{0}\eta {e}^{+}{\nu }_{e}$ and ${D}^{+}\to \eta \eta {e}^{+}{\nu }_{e}$ and we set the upper limits on their branching fractions.