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.

Dear Gyanendra,

in order to hopefully help you in most efficient way, I have copied your answer below and will comment it paragraph after paragraph. In general (and at this point) I have no specific experience/knowledge about photoinduced currents and their decay, apart perhaps from my own work on persistent photoconductivity in amorphous Ge films (published in J.Phys C, together with Gerhard Fasol). Therefore take my comments as general ones, regarding the electrical response in condensed phase :

Gyanendra Bhattarai · University of Missouri - Kansas City

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.

Petr: That I understood, but what is 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.

Petr: General comment :

The term "displacement currents" should be used for the time dependent electrical response of the bound charges, where the mobility is zero. The response of this type affects the dielectric response function eps,

The time dependent response you are talking about is due to charges with finite mobilities  and I named this response  "mobile charge polarisation" . This type of response does not change the dielectric function/constant eps, but rather the measured time(frequency) dependent resistance(real part of the measured electrical impedance), determined by mobile charge densitiy and mobility at place x and time t .

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.

Petr: I think you are right, but in general it will depend on how much extra charge (el-hole pairs) is generated at the left electrode (anode) and how much of this extra charge dies away on the way to the cathode. Using spacers as mica is quite a dangereous think to do because the overall response can be dominated by the time dependent effects at electrode mica and mica-sample interfaces. Finally I do not know what is Schockley-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).

Petr: Yes, but remember that this type generation creates extra charge many microns inside the sample, it is NOT a delta function type generation as is the case in electron beam generation (penetration depth is in Angstroms, well almost)

So in ideal case, the mechanism of decrease of carriers as they move forward is because of trapping.

Petr. No, it is because of them DISAPPEARING at the recombination centres (in electron and hole, out nothing !!). The recombination centres can be considered as true sinks and sources , charges disappear from the electrical circuit and/or are generated there.

The exponential decrease is not related to exponential attenuation of light intensity.

Petr: OK, that is good that we have clarified this point (at least in principle)

The paper assumes that the photogenerated carriers are low enough to believe the electric field inside is uniform (not affected by generated charges)

Petr: That is a fundamental fallacy !! Until you really calculate it, you can not make this assumption. The authors are apparently not aware that the depletion regions are in general of the same magnitude (the fields due to depleted regions might be (in fact they usually are !!) comparable to field changes caused by the external photo-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:

Petr: No, Although the displacement currents vanish, there will be a steady state current (electron plus hole) that is determined by the rerspectrive drift AND concentration gradients (it is actually given at all times by the gradient of the respective quasi-electrochemical potential(s) at a given quantum energy level(s) that contributes to the steady state electrical current.

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.

Petr : The traps are not filled uniformly in space due to the difference between the respective electro-chemical potential and the energy level in question. The electric field is never uniform in usual Schottky contacts. Only when the electrochemical potential of the metal electrode is identical/equal to electrochemical potential of the sample, the electric field inside the sample can be considered as uniform. This is of course never the case, it would require that the metal electrode and the sample are identical materials !

The carrier concentration is not spatially uniform in general. In general, when the sinks and sources are present, majority carrier concentration in x decreases and minority carrier concentration increases correspondingly. With no sinks and sources (deep levels - recombination centres) and in strong one carrier system (n or p), it is though correct that the majority carrier current is almost equal to the steady state total current through the sample. THIS DOES NOT MEAN that both the concentration and the field are uniform. On contrary, they are not, otherwise you would never get into steady state condition (constant, time independent current through the sample) !!!  

Continuity equation is an integral part of one of Maxwell equations and when sinks/sources are present, continuity equation IS NOT satisfied for each relevant quantum  energy level (each transport "channel").

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.)

Petr: Well, this is more of a speculation. Although I do this type of numerical calculations, I will not come up with such a strong argument/statement. The result will depend on the concentration of excess charges, their mobility and first of all on the capture cross sections (the probability that a moving charge will be captured by a trap and/or deep level(recombination centre). With almost zero capture cross sections, the carrier may be first trapped at the anode or not al all. Lastly, I do not see, why the x dependence of the excess charge in steady state should be exponentially decreasing ?!

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.

Petr: Again, this is a strong approximation, not valid in general. As you just  pointed out, the excess charges will create a space charge (non-zero field) and therefore there will (and must be !) always a term in the expression for the total current that is given by concentration gradient. For time->infinity(steady state),

J(x)= e*p(x)*mu(x)*E(x) - (e/kT)*D*dp(x))/dx

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.

Petr: If the equations to be solved are properly defined, then this is possible, because steady state=transient state(time->infinity)

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). 

Petr: The proper sequence is:  Start of the time->calculation of local, space-time dependent current density i(x,t) [Amp/M*2] ->calculation of the total current density through the sample at time t I(t)= integral(I(x,t)dx - > Next time tnew=timeold plus delta time ->back to the start, where time=time new. After sufficient time interval (in general time->infinity), I(t)->const and also (in fact the definition of the steady state) i(x,t)->i(x) = constant for all x. So,in the steady state, i(x) is the (must be) the same at each x, right from the anode to the cathode.

In short, I can say that, I don't understand how Shockley-Ramo theorem is used in steady state.

Petr: As I already mentioned, I do not know what Chockley-Ramo theorem states and therefore I can not give an useful comment.

I don't see its applicability as there is no displacement current unless there are electrons inside the material so the charge carriers disappear

Petr: You contradict yourself here, with trappnig hypothesis, you will get a loss of excess charges (disappearing into traps->no further movement) in space and time. But you are right that in the limit time->infinity(steady state), there will be (principle of detailed balance) no further time change in the current.

because of recombination

Petr: Yes

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.

Petr: The electrons do not have to move, they just have to be there ! As pointed by A.Kumar, there will always be a finite concentration of thermally excited electrons and ,in case of presence of deep levels(recombination centres) much large concentration of electrons there (only~half band gap activation energy). 

Best regards

Petr

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Gyanendra Bhattarai  

University of Missouri - Kansas City

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