Theory of steady-state plane tunneling-assisted impact ionization waves

The effect of band-to-band and trap-assisted tunneling on the properties of steady-state plane ionization waves in $p^+$–$n$–$n^+$-structures is theoretically analyzed. It is shown that such tunneling-assisted impact ionization waves do not differ in a qualitative sense from ordinary impact ionization waves propagating due to the avalanche multiplication of uniformly distributed seed electrons and holes. The quantitative differences of tunneling-assisted impact ionization waves from impact ionization waves are reduced to a slightly different relation between the wave velocity $u$ and the maximum field strength $E_M$ at the front. It is shown that disregarding impact ionization does not exclude the possibility of the existence of tunneling-assisted ionization waves; however, their structure radically changes, and their velocity strongly decreases for the same $E_M$. A comparison of the dependences $u(E_M)$ for various ionization-wave types makes it possible to determine the conditions under which one of them is dominant. In conclusion, unresolved problems concerning the theory of tunneling-assisted impact ionization waves are discussed and the directions of further studies are outlined.