Hot electron cooling in Dirac semimetal Cd$_3$As$_2$ due to polar optical phonons

A theory of hot electron cooling power due to polar optical phonons $P_{\rm op}$ is developed in three-dimensional Dirac semimetal($3$DDS) Cd$_3$As$_2$ taking account of hot phonon effect. Hot phonon distribution $N_q$ and $P_{\rm op}$ are investigated as a function of electron temperature $T_e$, electron density $n_e$, and phonon relaxation time $\tau_p$. It is found that $P_{\rm op}$ increases rapidly (slowly) with $T_e$ at lower (higher) temperature regime. Whereas, $P_{\rm op}$ is weakly deceasing with increasing $n_e$. The results are compared with those for three-dimensional electron gas ($3$DEG) in Cd$_3$As$_2$ semiconductor. Hot phonon effect is found to reduce $P_{\rm op}$ considerably and it is stronger in 3DDS Cd$_3$As$_2$ than in Cd$_3$As$_2$ semiconductor. $P_{\rm op}$ is also compared with the hot electron cooling power due to acoustic phonons $P_{\rm ac}$. We find that a crossover takes place from $P_{\rm ac}$ dominated cooling at low $T_e$ to $P_{\rm op}$ dominated cooling at higher $T_e$. The temperature at which this crossover occurs shifts towards higher values with the increase of $n_e$. Also, hot electron energy relaxation time $\tau_e$ is discussed and estimated.