I have read a paper about an application of modulation doping in bulk thermoelectric materials. In this paper, the nanoparticles diffuse carriers into the host materials, and I want to estimate the diffusion length from nanoparticles to the host.
I know the formula \$D=\sqrt{D\tau}\$, where \$D\$ is the diffusion coefficient and \$\tau\$ is the life time of carriers. But in modulation doping, the carriers in the host do not have recombination, and I don't know how to get the life time and diffusion length.
I have tried to simplify this problem to an electromagnetic problem, and I got a too small diffusion length to believe. I used the formula: \$E=\frac{Qe}{4πϵ}(\frac{1}{R}−\frac{1}{a})\$. \$E\$ is the band offset between materials of host and nanoparticle, \$a\$ is the radius of the nanoparticles, \$R-a\$ is the diffusion length, and \$Q\$ is the amount of electricity in a nanoparticle.
How do I estimate the correct length?
