I asked this question before and explained it in the body of the question analysis. In this question I would like to share one perspective where tunneling based transport might oversee drift based transport and render Ohm's law inapplicable in this regard. For example, for atomically short device length, the field is very high such that slope of Ec and Ev are almost perpendicular or very steep. Also the carrier number will be minute as the atomic volume is very small. The doping states close to conduction band will also more or less get align as per EC and EV slope. As a result the Ec level electron is directly in-line with a vacant doping energy state where it can tunnel back. Then from that state it can tunnel back to Ec level and since the extension of Ec in the length direction is in atomic or nm dimension, chances are that within couple of back-to-back tunneling electrons can propagate from cathode to anode of the material where carrier drift might not be central mode of transport and hence Ohm's law might not be appropriate here.
Dear Nabil,
I don't know exactly what is your perspective, however, it sounds like explanation for Esaki-diode.
Dear Yoshiki:
Regards.
Apart from Esaki diode, in a single material acting as a resistive device whose one end is anode, the other is cathode and the material is atomically long, such that V/L is significant, please check my question where the slope of Ec and Ev will be also very large and almost perpendicular considering L in nm. Then instead of drift that governs Ohm's Law, tunneling between aligned energy states of conduction band electron and vacant donor energy states are possible and in this way they can be transported from cathode to anode terminal since L is in nm which may violate Ohm's Law.
Sincerely,
Dr. Nabil Shovon Ashraf.
Dear Nabil,
I don't understand well. You mean "single material" as n-type only? You mention about vacant donor state. It's energy level is normally close to the Ec, hence tunneling between Ec and the state is almost same with drift. Can you show a picture of the band structure?
Yoshiki:
I do not need to explain it anymore to you since you guessed it right. Let's suppose a single n-type material with nano dimension. Ec has a steep slope as determined by V/L and donor states will align with as much as same slope like Ec. Then you guessed it right that there is almost same probability that during drift some electrons will tunnel into these vacant donor states and get reemitted into conduction band by tunneling back. But tunneling has a different current-electric field relationship where as for drift, current is linearly related with electric field. If we include tunneling in transport process of electrons from cathode to anode by the process I described and you somewhat have the picture, then Ohm's law might turn out to be invalid for this extremely nm dimension n-type resistive material built in a semiconductor.
Sincerely,
Dr. Nabil Shovon Ashraf
Dear Nabil,
In my understanding, a tunneling occurs when the energy levels are same. In your perspective, electrons can not travel from the cathode to the anode.
I agree with that Ohm's law is no more valid. In my view, electrons jump in the anode with energies of the cathode level---we call them "hot carriers". Is it really happen? I disagree. Because there are many factors to be considered. The band structures of the contacts are big concerns. To see hot carrier phenomenon in the reality, a heterostructure of semiconductors seems better. I think there are some studies. I was actually studying about possibility of hot carrier solar cells. It was years ago.
Yoshiki:
I take your point but wanted to give some more light on my view. During tunneling electrons are at the same energy level but what I said is during drift some electrons might undergo tunneling. Therefore the aggregated transit time of carriers between anode and cathode contact will be delayed compared to the case if the transport is purely drift based and a phenomenon like nonstochastic transport as a result of tunneling might seem a possibility. Fluctuations in the current may also result and a constant slope between applied voltage and current-a characteristic of a resistor will not happen invalidating Ohm's law.
Sincerely,
Dr. Nabil Shovon Ashraf.
How about the dependence of tunneling current on the size of contact surface?
It is also non-Ohmic.
I agree with your comment Dr. Eugene.
What it proves in nm single resistor is that drift transport may be interrupted by tunneling between conduction band carriers aligned with dopant energy states and as a result of which current will be fluctuating due to non-uniform conduction band carriers involved in the drift and the slope of voltage-current will be a non constant value from point to point spatially as we take the whole length of the resistor. Also contacts may play a role here.
Sincerely,
Dr. Nabil Shovon Ashraf.
May be You know any reference about tunneling current dependence on contact aria, Nabil?
Schottky barrier contact is one source of tunneling current from the contact, it happens when the source is not too degenerately doped. If both contact region and the body of the bulk material are degenerately doped, the ohmic nature of the contact might not be established due to nm device size and fabrication issue, in this case band-to-band tunneling is possible since both contact and body are degenerately doped. I read couple of excellent papers on this issue.
Sincerely,
Dr. Nabil Shovon Ashraf.
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