You can not create electron- hole pair in a perfect insulator by light. Definitely you must be dealing with a semiconductor. CdS is a semiconductor where electron hole pairs can be generated by light shining.Every semiconductor will also have thermally generated charge carriers which will be equal in number in intrinsic case. When you create excess charge carriers, electron hole pairs, at one end , i.e. at anode, holes will move towards the cathode they can not be recombined by traps as traps temporarily captures the charge carriers and then release. Recombination will be with thermally generated electrons only.

Well, I do not agree. You can generate electron-hole pairs in any material, the question is just how many and whether your light has sufficient energy. Time of flight measurements of drift mobilities in silica (band gap ~8eV !) is a clear example that e-h can be generated, if not by light then by electron beams.

In your case, and here I agree with A.Kumar,  the situation is probably controlled by thermal electrons in conduction band (excited from valence band or from shallow impurities-donors) , but even more likely by deep levels within the gap, where the thermal equilibrium concentration of electrons is orders of magnitude larger. These then act as recombination centres, not as traps for one or the other charge carrier.  

My last comment concerns the life-time of excess carriers in very pure and defect free systems such as mono-crystalline, ultra pure Silicon for example. Here, the lifetimes in excess of 10 miliseconds are typical, leading to several traverses of a given charge between the electrodes !!  

And to your specific points :

1. You get always electron/hole "injection" (a bad term, transfer is a better one) into your sample, the amount being determined first of all by the difference in electrochemical potential of used metal electrode and electrochemical potential of your sample. I would suggest that you check again that the exponentially decreasing hole concentration with distance into the sample is not just result of the electron-hole generation process (the light gets absorbed within the sample in this fashion - I=Io*exp(-alpha*x). Such would be a profile of the excess hole concentration in the final steady state.l

2.   As mentioned above, the only possible recombination process are the direct conduction band to valence band (the finite concentration of thermal electrons at the bottom of conduction band) and/or recombination via deep levels (mid-gap states). Otherwise, the excess charges life time will be infinite (well, let us say seconds in a real material, since even Silicon has residual, unwanted defects and impurities)

3. If recombination via deep levels is present, then both currents (electron and hole) must be calculated, since while in one part the current is predominantly hole-like, in the other part it is predominantly electron-like. See also the book by de Boer: Electron transport in Semiconductors (Wiley 2002). So, you are right, there is electron current contribution, otherwise the condition of a steady state (constant current through the sample) would not be satisfied.

I hope this will help you on the way

With regards

Petr

Thank you Petr Viscor for your time and answering my question.

Regarding my question,  I was trying to understand what mechanism controls the current in steady state. Although not explicitely stated, my intention was to understand the use of steady state photocurrent for Hecht analysis.

In this context, I have now a bit different question. Though it is quite long, I hope you take a look.

1.  In the paper I cited above, "High field effects in Photoconducting CdS"  Many talks about the transient current induced (displacement current) by the photogenerated charge moving inside CdS crystal. Clearly, they can not make steady state photocurrent as the crystal is not connected electrically. They are using mica spacers to couple the crystal capacitively. I believe the displacement current is same as the current explained by shockley-ramo theorem.

2. In one case, they use the approximation of surface absorption of light to create holes (they irradiate +ve contact with light so holes gererated migrate towards negative contact). So in ideal case, the mechanism of decrease of carriers as they move forward is because of trapping. The exponential decrease is not related to exponential attenuation of light intensity. The paper assumes that the photogenerated carriers are low enough to believe the electric field inside is uniform (not affected by generated charges)

Below is my understanding and confusion.

Understanding:  

Suppose we have direct electrical contact say blocking so we can measure steady state photocurrent. In steady state, I think the displacement current contribution vanishes and the continuity equation is explained only by drift current. If there are traps, I think two possibilities:

a. The traps are filled uniformly spatially and so the electric field remains uniform. The drifting carrier concentration remains uniform and so continuity equation is hold.

b. Since carriers are generated near anode, they are trapped more in anode region and so the carrier density drops exponentially along distance. In this case, for the continuity equation to hold, the space charge is created and the electric field will be non uniform (like in space charge limited current.)

However in both cases, the total current is because of drift current and the total current is given by J = e*p(C)*mu*E(C). where p(C) is density of holes collected at cathode and E(C) is electric field at cathode. I don't think we need to integrate the charge carriers entire sample to get the current.

Confusion:

There are many papers describing  Hecht analysis. Most of them cite Many's paper. Though Many's paper describes transient current, other authors use the same mathematics for steady state. Though they find p(z), E(z) for every z, they  integrate  e*p(z)*mu*E(z) over the entire length and divide by the volume of sample to get the current. I am not convinced by this integration because once one knows p(z), E(z), the current is just J =  e* p(z) *mu* E(z) = e* p(C)*mu* E(C). 

In short, I can say that, I don't understand how Shockley-Ramo theorem is used in steady state. I don't see its applicability as there is no displacement current unless there are electrons inside the material so the charge carriers disappear because of recombination which then requires S-R theorem to calculate total current. But then, the current will not be entirely due to holes. There must be some way that electrons move within the crystal in CB to recombine with holes.