Ballistic transport in a semiconductor with collisions

At low temperature in a high mobility semiconductor such as GaAs the transit time of electrons across a short device may become comparable or even less than the time between collisions. Under such conditions electrons may move almost ballistically accelerating in the electric field. Such a situation is analyzed in the frame of a simple model assuming a single energy independent relaxation time. This model allows solving a problem in an analytical form. The model predicts that at high injection level (voltages larger than a punchthrough voltage) the
I-V
characteristics change from the case described by the Child-Langmuir law for a short device and/or a large time between collisions to the case described by the Mott-Gurney law in the collision-dominated case. At low injection level the space oscillations of the electric field with the wavelength
\lambda = 2\pi\v/\omega_{p}
may appear (ω
p
is the plasma frequency and
v
is the electron velocity) and the
I-V
characteristic may become multivalued (S-type) due to the influence of the positive donor charge leading to the space overshoot of electrons.