Effect of carrier drift velocities on measured ionization coefficients in avalanching semiconductors

We point out that, contrary to what was previously assumed, the macroscopic ionization rates defined through the equations for the current densities and determined from multiplication data in $p-n$ junctions, are generally different from the microscopic ones (i.e., the inverse of the average distance traveled by an electron or a hole between ionizing collisions). This difference is due to the possible field dependence of the carriers' drift velocities in the avalanche regime, and is proportional to the field gradient in the $p-n$ junction. We have estimated this effect in Si, where experimental data seem to indicate a gradual increase of the drift velocities above the scattering-limited values in the field range $2\times {10}^5 \mathrm{V}/\mathrm{cm}\le E\le 5\times {10}^5 \mathrm{V}/\mathrm{cm}$, in agreement with existing theories. In one-sided Si $p^{+}-n$ abrupt junctions for $n\simeq 1.5\times {10}^{16}$ ${\mathrm{cm}}^{-3\mathrm{}}$ and for a maximum field $E_m\simeq 3.5\times {10}^5$V/cm, we estimate a correction of ≈ 10% (50%) on the electron (hole) ionization rate, which rapidly decreases with increasing field. Our analysis suggests the convenience of using a $p-i-n$ geometry for a direct determination of the microscopic ionization rates. For a $p-n$ junction, it is necessary to know the field dependence of the drift velocities in order to fit the current microscopic models to the measured ionization rates.