Hot electron energy relaxation in lattice-matched InAlN/AlN/GaN   heterostructures: the sum rules for electron-phonon interactions and   hot-phonon effect
  Using the dielectric continuum (DC) and three-dimensional phonon (3DP) models, energy relaxation of the hot electrons in the quasi-two-dimensional channel of lattice-matched InAlN/AlN/GaN heterostructures is studied theoretically. The electron power dissipation and energy relaxation time due to both half-space and interface phonons are calculated as functions of the electron temperature $T_e$ using a variety of phonon lifetime values from experiment, and then compared with those evaluated by the 3DP model. The 3DP model yields very close results to the DC model: with no hot phonons or screening the power loss calculated from the 3DP model is 5% smaller than the DC power dissipation, whereas slightly larger 3DP power loss (by less than 4% with a phonon lifetime from 0.1 to 1 ps) is obtained throughout the electron temperature range from room temperature to 2500 K after including both the hot-phonon effect (HPE) and screening. Very close results are obtained also for energy relaxation time with the two phonon models (within a 5% of deviation). However the 3DP model is found to underestimate the HPE by 9%. The Mori-Ando sum rule is restored by which it is proved that the power dissipation values obtained from the DC and 3DP models are in general different in the pure phonon emission process, except when scattering with interface phonons is sufficiently weak, or when the degenerate modes condition is imposed, which is also consistent with Register's scattering rate sum rule. Our calculation with both phonon models has obtained a great fall in energy relaxation time at low electron temperatures ($T_e<$ 750 K) and slow decrease at the high temperatures. The calculated temperature dependence of the relaxation time and the high-temperature relaxation time $\sim$0.09 ps are in good agreement with experimental results. 
Jian-zhong Zhang, Angela Dyson and Brian K. Ridley
cond-mat.mes-hall