It is well known that conductivity of insulator increases with electric field because of increased carrier concentration by Poole-Frenkel emission. There is a renowned paper by Murgatroyd dealing with Poole-Frenkel emission with space charge limited current.
What confuses me is, if Poole-Frenkel emission starts before the onset of space charge, does not it change the onset voltage of space charge because of increased conductivity/concentration? Or in some case, does not it make space-charge effect not to take place at all? The problem is described below in detail.
In simple understanding, the space-charge starts when the injected charge becomes greater than the thermal equibilibrium charge in material, mathematically,
CV > n0edA ; C is capacitance, V is applied voltage, n0 is concentration, d is thickness and A is area of device. This can be written as
E>n0ed/epsilon; E is electric field, epsilon is permittivity of material ( I assume permittivity is not much dispersive).
Now suppose that Poole-Frenkel emission starts before space charge. The concentration keeps increasing with the applied field (I don't say that it increases infinitely as it saturates based on trap density). The concentration after including PF is given as
n = n0 exp(\beta\sqrt(E)); beta is PF coefficient and at 300K for a material of dielectric constant 5, it is 0.001319 (m/V)^1/2.
The condition for space-charge to start becomes
E > (n0 e d/epsilon) exp (\beta\sqrt(E))
rearranging,
d < (epsilon/n0*e)*E* exp(-\beta\sqrt(E))
This equation, for a material with n0 = 10e14 at 300K, epsilon = 5, gives the maximum thickness of sample around 850 nm.
Suppose we have a 500 nm thick film. In low field, the field itself is not sufficient to produce space charge. At around 4e3 V/cm, the sample can reach to space charge and reamin in space charge regime up to 8.5e4 V/cm. Beyond that the exponential increase in conductivity (concentration) by PF makes it hard to bring the device in space charge. so the conduction should fall back to Poole-Frenkel only.
The field of 8.5e4 V/cm increases the carrier concentration by 47 times. There are papers on high field saturation of PF effect but the field is not so large, so the number 47 might be possible (unless the trap density is very small and so not enough carriers are left in trap to be excited by the applied field.)
If this is the case, then how do we get thick devices into space charge regime?
Any help in this is appreciated.
Dear Gyanendra,
thanks for the help, now I can see what you are asking about.
Not an easy question to answer straight away. In order for me to say something useful, I need some more information :
1. Is your question general (theory/model of space-charge limited current) or do you have a specific system you want to investigate. If the latter is the case, please describe your experimental sample and conditions of measurement.
Right now, I haver following comments :
2. Depending on the system under investigation, there are two different situations to be considered:
The first is, when the electrical transport is considered in the system, where the transport is via hopping of charges from one localised state to the next, as is the asituation in glasses and amorphous thin films (see the book by Mott-Davies: Electronic processes in non-crystalline materials (p.97,269,490). Here the Poole-Frenkel effect affects NOT the concentration, but the MOBILITY of the charge through the change in the hopping rate.
The second is when the mobility of the transport through the localised states is zero and the system conducts electricity by injecting charges from the left electrode and collecting them at the right electrode. The charges move by trapping and de-trapping at and from some postulated deep traps. Here it is more the de-trapping rate (thereby the concentration of charges) that is rate-limiting process. In fact , and in a more modern approach, it is the same process as with the effect of deep levels in crystalline semiconductors, just with the background charge density zero.
3. The Poole-Frenkel effect changes therefore either the hopping rate or the de-trapping rate through lowering the assumed energy barrier.
4. I, myself, I would not spend too much time in trying to analyse in detail these types of approach (space-charge limited current equation, Mott-Gurney equation, Child-Langmuir equation, and possibly others,...) since all of them are some approximations to the real physical world. Here, I will always advocate another approach, namely Electrical Impedance Spectroscopy approach, where the various above mentioned equations (valid ONLY at various limited regions of applied dc voltage) come out naturally and are only functions of the fundamental electrical material parameters of the System Under Study (SUS).
5. Because of point 4. , I have never (up till now) got down to see precisely what "the space-charge limited current" means in real world of real physical systems, either mono-crystals, glasses and/or liquid electrolytes.
It might be that this will help, otherwise I need some more input from you concerning the specific problem at hand. At some point, I will probably do some analysis, relating the dc-type electrical measurements like space-charge limited transport to the more general ac transport, but it is not my intention to do this right now, unless you persuade me otherwise.
With best regards
Petr
Thank you Petr for the answer.
1. Though I am working to find charge carrier mobility of very high resistivity / low mobility (expect it to be around 1e-8 -1e-6 cm2/Vs) amorphous thin films, this particular question is about theory in general for crystalline insulators with traps/donars so the PF emission increases the concentration, not the mobility (though this effectively may change the mobility as well).
2. Since my system is amorphous, I am aware of the field-dependent mobility of PF type given as mu = mu_0 exp(\gamma\sqrt(E)), with gamma both positive/negative depending on the temperature and spatial disorder. For crystalline material however, the concentration may matter. What I really don't like (or don't understand) is researchers use a particular approximation based on what they are trying to see. For example, In a paper "Dielectric breakdown and Poole-Frenkel field saturation in silicon oxynitride thin films' by S Habermehl and R.T. Apodaca, they are doing I-V measurements up to 10MV/cm of field for film thickness less than 50 nm. Though they don't have concentration reported, I assume the field is enough to build space-charge. However, they do not talk about it at all. If I was doing this experiment , I would use Murgatroyd expression of SCLC current with PF emission. Other people might have even different perspectives. (Specially about the paper, please let me know if my understanding is wrong).
4. At a point, I was also doing Impedance spectroscopy measurements for different study. However, particularly for mobility extraction, only the method I am familier with is the decreased capacitance at low frequency because of the transit time effect of injected carriers by small ac singnal in space-charge created by APPLIED DC BIAS. The problem is same again, we need to have the device in SCLC regime and... No luck. This works for a particular set of experimental conditions (thickness, available bias source, how much external bias the impedance analyzer can support, and of course, the mobility of material.
If you can suggest any alternative method of determining mobility using IS, I would be really greatful as I spend a long time in making understanding of IS (though I don't know much :))
thank you very much for your comments and explanations.
Gyanendra.
Dear Gyanendra,
I have the following comments that will be hopefully useful :
1. I agree that when the injected charges get trapped and de-trapped, it is the finite charge concentration that is created (not increased, since in a perfect insulator, the intrinsic mobile charge density is zero) through the sample by injection. In most simple case and under steady state conditions (if attainable !), the sample gets, so to speak, charged-up homogeneously (in space, from the right electrode to the left electrode). If the system is characterised by "Ohmic" contacts, there will be no extra accumulation(depletion) of the charge within the electrode-sample interface region. All of this though, will happen also at zero dc bias ! Poole-Frenkel effect will "only" increase the de-trapping rate. It is appropriate here to define what is meant by "space charge" or by "space charge region". It is a spatial region of the sample, where the electrical charge neutrality is broken and the electrical charge density ro(x,t) is non-zero. The effect of such regions within the sample on the flowing steady state dc current is not an easy and straight forward task to solve and in general, numerical simulations are needed. In this general sense, all the systems where there are either free surfaces and/or metal electrodes attached, there will be space charge region(s).
2. I have not read the paper by S.Habermehl, but I could envisage a situation, where the dc current density is "Ohmic" also at higher values of applied dc bias. From the numbers you quote, they apply only 10 volts across their film (1 MVolt/cm), so it might be that they are justified in not considering Poole-Frenkel effect.
3. Determination of the mobility. The method you refer to (change in capacitance at lower frequencies) is the change of the depletion space charge region width near contacts with increasing applied dc bias. It is known as Mott-Schottky method (plot) and when this particular capacitance is plotted as 1/C^2 versus dcbias , you can extract density of the mobile charges from the slope. I am not aware (in fact I do not agree !) that this method can be used for mobility extraction. It is not the capacitance level that changes with changing mobility, it is the frequency region, where the bulk capacitance changes towards the depletion capacitance that makes such a determination possible. This analysis is precisely one of the unique features of the Electrical Impedance Spectroscopy (EIS). It is the only experimental method capable of determining both the mobility and the density of the mobile electrical charges simultaneously and in one experiment. Other methods (Hall effect+dc conductivity measurements, Time-Of-Flight measurement and possibly others) are either inadequate, can not be used (Hall effect in amorphous systems) or are difficult to analyse (dispersive transport in TOF experiment used on amorphous samples). So, short answer to your question is that yes, EIS can in principle be used to determine both the mobility and density of your mobile chargers. In your situation I would strongly consider purchase of a good Impedance Analyzer. For me to see, it is the only way how you can solve your task of finding mobility and the density of mobile charges in your amorphous thin films.
With best regards
Petr
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