3D Time-of-flight distance measurement with custom - Universität ...

SOLID-STATE IMAGE SENSING 63
transport if many free charge carriers are located close to each other. We will see
later that for typical optical conditions in our TOF application, we deal with the
transportation of single electrons. Therefore, self-induced drift is neglected.
Self induced drift Thermal diffusion
E
Electric field
Figure 3.11 Transport mechanisms for free electrons in a semiconductor.
Thermal diffusion
For temperatures larger than T=0°K, free charge carriers have thermal energy,
which results in a permanent movement in random directions. The thermal velocity
vth, i.e. the amount of the microscopic movement velocity, is given by:
3kT
v th =
m
Equation 3.6
eff
At room temperature (300K) we get vth≈10 7 cm/s for Si and GaAs. The charge
carriers move fairly quickly but steadily change their directions due to collisions with
the fixed lattice or impurity atoms or other free charge carriers. The sum of these
microscopic movements results in a macroscopic movement, again in random
directions. This movement process, known as thermal diffusion, only ends when the
free carriers recombine after a certain lifetime τn and τp respectively. Statistically
speaking, after this time they have traveled a certain distance, the so-called
diffusion length Ln:
Ln
= Dn
⋅ τn
Equation 3.7
This diffusion length depends on the doping concentration of the semiconductor.
This is expressed in the diffusivity Dn, which contains the doping dependent carrier
mobility (electron mobility µn):