Electron-Acoustic Phonon Energy Loss Rate in Multi-Component Electron Systems with Symmetric and Asymmetric Coupling Constants

We consider electron-phonon ($\mathit{\text{e-ph}}$) energy loss rate in 3D and 2D multi-component electron systems in semiconductors. We allow general asymmetry in the $\mathit{\text{e-ph}}$ coupling constants (matrix elements), i.e., we allow that the coupling depends on the electron sub-system index. We derive a multi-component $\mathit{\text{e-ph}}$power loss formula, which takes into account the asymmetric coupling and links the total $\mathit{\text{e-ph}}$ energy loss rate to the density response matrix of the total electron system. We write the density response matrix within mean field approximation, which leads to coexistence of
symmetric energy loss rate $F_S(T)$ and asymmetric energy loss rate $F_A(T)$ with total energy loss rate $F(T)=F_S(T)+F_A(T)$ at temperature $T$. The symmetric component F_{S}(T) $ is equivalent to the conventional single-sub-system energy loss rate in the literature, and in the Bloch-Gr"{u}neisen limit we reproduce a set of well-known power laws $F_S(T)\propto T^{n_S}$, where the prefactor and power $n_S$ depend on electron system dimensionality and electron mean free path. For $F_A(T)$ we produce a new set of power laws F_{A}(T)\propto T^{n_{A}}$. Screening strongly reduces the symmetric coupling, but the asymmetric coupling is unscreened, provided that the inter-sub-system Coulomb interactions are strong. The lack of screening enhances $F_A(T)$ and the total energy loss rate $F(T)$. Especially, in the strong screening limit we find $F_A(T)\gg F_S(T)$. A canonical example of strongly asymmetric $\mathit{\text{e-ph}}$ matrix elements is the deformation potential coupling in many-valley semiconductors.