Intervalley-Scattering-Induced Electron-Phonon Energy Relaxation in Many-Valley Semiconductors at Low Temperatures

We report on the effect of elastic intervalley scattering on the energy transport between electrons and phonons in many-valley semiconductors. We derive a general expression for the electron-phonon energy flow rate at the limit where elastic intervalley scattering dominates over diffusion. Electron heating experiments on doped $n$-type Si samples with electron concentrations $(3.5–16.0)\times {10}^{25}{\mathrm{m}}^{-3}$ are performed at sub-Kelvin temperatures. We find a good agreement between the theory and the experiment.

Figure 1

Schematic illustration of the constant energy ellipsoids of Si conduction band valleys. The valleys are located close to the $X$ point in the first Brillouin zone. Elastic scattering rates $1/{\tau }_1$ and $1/{\tau }_2$ couple the different classes of the valleys.

Figure 2

(a) Schematic illustration of the sample geometry and the measurement setup. The $\sim 9500\mu \mathrm{m}$ long $n^{+}$ Si film is heated with a dc current density $J$. $T_e$ and $T_{\mathrm{ph}}$ are measured using current biased S-Sm-S (Al-Si-Al) contacts (only the biasing circuit for $T_e$ is depicted). The $T_{\mathrm{ph}}$ thermometer is electrically isolated from the main Si film by a $\sim 1\mu \mathrm{m}$ gap. (b) The power density $P={\rho }_e{J}^2$ vs $T_e^6-T_{\mathrm{ph}}^6$.

Figure 3

Slopes $S$ of the linear fits in Fig. 2b and intervalley scattering time ${\tau }_2$ [determined from $S$ and Eq. (7)] as a function of electron density $N$. The dashed curve is a polynomial fit that serves as a guide for the eye. The inset shows tabulated values of $S$ and ${\tau }_2$ (in the units of the axes).