Diffusion of charge carriers

[MathematicalPhysicist]

Homework Statement

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?

Homework Equations

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.

 

[mark726]

Hello,

Thank you for sharing your thoughts on this problem. I am a scientist and I would like to contribute to this discussion.

For question 1, I agree with your answer. After Debye time, the densities will decrease exponentially in space due to diffusion. This is due to the fact that the initial distributions are Gaussian, which is characteristic of diffusion processes.

For question 2, we can use the diffusion equation to find the long-time dependence of the densities. The diffusion equation is given by:

$$\frac{\partial \Delta n}{\partial t} = D_c \nabla^2 \Delta n$$

where $D_c$ is the diffusion constant for electrons. We can solve this equation for $\Delta n(\vec{r},t)$ by using the method of separation of variables. The solution will be of the form:

$$\Delta n(\vec{r},t) = \sum_{n=0}^{\infty} A_n e^{-D_c \lambda_n t} \phi_n(\vec{r})$$

where $\lambda_n$ and $\phi_n$ are the eigenvalues and eigenfunctions of the diffusion equation, respectively. The long-time dependence of the densities will depend on the values of $\lambda_n$. For example, if $\lambda_0=0$, then the long-time dependence will be constant. If $\lambda_0<0$, then the long-time dependence will be exponential decay.

To find the values of $\lambda_n$ and $\phi_n$, we can use boundary and initial conditions. In this case, the boundary condition is that the densities must go to zero at infinity. The initial condition is given by the initial distribution. This will lead to a set of equations that can be solved numerically to find the values of $\lambda_n$ and $\phi_n$. Once we have these values, we can plug them into the solution to find the long-time dependence of the densities.

I hope this helps. Let me know if you have any further questions.