Steady state of diffusion current in semiconductors

1. Apr 26, 2014

Anupam

Consider a long semiconductor bar is doped uniformly with donor atoms so that the concentration is given by n = ND 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 ip 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.

I wish to determine the time it takes for this system to reach steady state, that is the general expression. I know it takes infinite time to actually reach the steady state. I want to calculate say in how much it reaches 90% to the steady state.

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 τp is the mean life time, from the definition of mean life time and assuming that τp is independent of the magnitude of the hole concentration, p0 is the value of p in thermal-equilibrium value, g = p0p is the generation rate, p/τp 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 carrier concentration to reach this steady state value?

2. May 6, 2014