Steady state of diffusion current in semiconductors.

Question

Consider a long semiconductor bar is doped uniformly with donor atoms so that the concentration given by \(n = N_D\) and is independent of position. Radiation falls upon the end of the bar at \(x=0\), this light generates electron-hole pairs at \(x=0\). light keeps on falling.
          
Explanation: 

Because the material is n-doped (many electrons) the light does not significantly change the electron concentration. However, there are initially very few holes in the material, so the illumination does significantly change the number of holes. Holes in a n-type semiconductor are referred to as minority carriers.

Carrier transport in semiconductors takes place by drift and diffusion.  The hole drift current can be ignored (We shall make the reasonable assumption that the injected minority concentration is very small compared with the doping level.

The statement that the minority concentration is much smaller  than the majority concentration is called the low-level injection condition. Since the drift current is proportional to the concentration and we shall neglect the hole drift current but not the electron drift current  and shall assume that  \(i_p\) is due entirely to diffusion. This assumption can be justified (see e.g. Electronic Principles, Paul E. Gray & Campbell L. Searle, John Wiley & Sons 1969, or Millman's Electronic Devices).  The diffusion current density is proportional to the gradient in minority carrier concentration (in this case the holes) and diffusion coefficient,

$$j_p = -qD_p\frac{\partial p}{\partial x}$$
by Fick's law.
 

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Steady state is the state at which the parameters (e.g current density and carrier concentration) at a particular position \(x\) do not change with time. The continuity equation related to carrier current and generation and recombination rate is

$$\frac{\partial p}{\partial t} = -\frac{1}{q}\frac{\partial j_p}{\partial x} + G,$$

where \(\tau_p\) is the mean life time, from the definition of mean life time and assuming that \(\tau_p\) is independent of the magnitude of the hole concentration, $p_0$ is the value of $p$ in thermal-equilibrium value, \(g = p_0/\tau_p\) is the generation rate, \(p/\tau_p\) is the recombination rate, and \(G\) is the sum of generation rate and recombination rate.

Substituting the first equation and the value of \(G\) into the second gives

$$\frac{\partial p}{\partial t} = D_p\frac{\partial^2 p}{\partial x^2} + \frac{p_0 - p}{\tau_p}.$$

In the steady state \(p\) doesn't vary with time but vary w.r.t. position and the concentration at \(x=0 \)will remain constant all the time  hence we can put $$\frac{\partial p}{\partial t}=0;$$

hence when steady state is achieved we will have

$$\frac{\mathrm d^2 p}{\mathrm d x^2} = \frac{p - p_0}{D\tau_p}.$$

​How much time will it take for the minority carriers to reach the steady state concentration level?

Comments

It should be noted that it is not just about solving differential equations. The Fick's law is an approximation of ,\(\phi(x)=\dfrac{l_n}{2\tau}(n_1-n_2)\) where   \(n_2-n_2\) is replaced by \(-\dfrac{dn(x)}{dx}l_n\). I hope notations are understood. Also I have posted it from http://physics.stackexchange.com/questions/82949/steady-state-of-diffusion-current-in-semiconductors where the answerer did enough effort to do the maths and I respect his effort, though it doesn't seem to be fully correct.