# Diffusion of charge carriers

1. Apr 21, 2017

### MathematicalPhysicist

1. The problem statement, all variables and given/known data
You dope an $n$-type substrate at time $t=0$ with electrons and holes around the points: $\vec{r}_{0,c}$ and $\vec{r}_{0,v}$ respectively.

The initial densities' distributions are:

$$\Delta n (\vec{r},t=0) = \frac{\Delta N_0}{(2\pi a_{0,c}^2)^{3/2}}e^{\frac{-(\vec{r}-\vec{r}_{0,c})^2}{2a_{0,c}^2}}$$

where for $\Delta p(\vec{r},t=0)$ is the same as $\Delta n$ just interchange $a_{0,c} \to a_{0,v}$ and $\vec{r}_{0,c}\to \vec{r}_{0,v}$.

The questions are:
1. How will the densities look like after Debye time?
2. Find the dependence of the densities on long times, where the long time is compared to Debye time?

2. Relevant equations

3. The attempt at a solution
For question 1., what I thought is that the densities after Debye time will look like:
$$\Delta n(\vec{r},t)=\Delta n(\vec{r},0) e^{-|\vec{r}|/\sqrt{D_c\tau_D}}$$

The same with $\Delta n(\vec{r},t)$ just replace $n$ with $p$ and the diffusion constant $D_c$ with $D_v$.

If this answer is correct, I am still not sure how to answer question 2.

BTW, $\tau_D$ is Debye time, or the relaxation time.

2. Apr 26, 2017

### PF_Help_Bot

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