Crossover in the electron-phonon heat exchange in layered nanostructures

We study theoretically the effect of the effective dimensionality of the phonon gas distribution on the heat exchange between electrons and phonons in layered nanostructures. If we denote the electrons temperature by $T_e$ and the phonons temperature by $T_{ph}$, then the total heat power $P$ is proportional--in general--to $T_e^x - T_{ph}^x$, the exponent $x$ being dependent on the effective dimensionality of the phonon gas distribution. If we vary the temperature in a wide enough range, the effective dimensionality of the phonon gas distribution changes going through a crossover around some temperature, $T_C$. These changes are reflected by a change in $x$. On one hand, in a temperature range well below a crossover temperature $T_C$ only the lowest branches of the phonon modes are excited. They form a (quasi) two-dimensional gas, with $x=3.5$. On the other hand, well above $T_C$, the phonon gas distribution is quasi three-dimensional and one would expect to recover the three dimensional results, with $x = 5$. But this is not the case in our layered structure. The exponent $x$ has a complicated, non-monotonous dependence on temperature forming a "plateau region" just after the crossover temperature range, with $x$ between 4.5 and 5. After the plateau region, $x$ decreases, reaching values between 3.5 and 4 at the highest temperature used in our numerical calculations, which is more than 40 times higher than $T_C$.