Basically I have one or two very basic queries on temperature dependence of carrier concentration. As per book chapter 14 from the book "Physical Foundations of Solid-State Devices" & also from journal paper entitled "Shallow and deep donors in direct-gap n-type AlGaAs:Si grown by molecular-beam epitaxy" temperature dependence of carrier concentration for dual dopant energy level model (both donor type) at high temp can be described as:
nNDD+= (1/2)*NDDNcexp(-Edd/kT) [At high temp]
-> n(n-NSD)=(1/2)*NDDNcexp(-Edd/kT)
-> n2 - nNSD-(1/2)*NDDNcexp(-Edd/kT)=0 ......................(i)
where, NSD & NDD are the concentration for shallow & deep donor level respectively & Edd & Esd are deep & shallow donor ionization energy respectively.
One of the root of equation (i) will be n=[NSD + sqrt(2NDDNcexp(-Edd/kT))]/2
For high temp (T) regime,
exp(-Edd/kT) >> 1 & also NDD (Deep donor density) >> NSD (Shallow donor density)
Therefore, product of those above two: NDDNcexp(-Edd/kT) >> NSD2
-> sqrt(2NDDNcexp(-Edd/kT)) >> NSD
Thus root of equation (i) can be written as n = sqrt(2NDDNcexp(-Edd/kT))/2
= sqrt(NDDNc/2) exp(-Edd/2kT)
= n0 exp(-Edd/2kT) [at High Temp]
On the other hand, nNDD+ = (1/2)*NDDNcexp(-Edd/kT) can also be written as,
-> nNDD = (1/2)*NDDNcexp(-Edd/kT) [Since, at high temp, NDD+ = NDD as almost
all deep donor atoms become ionized]
-> n = (1/2)*Ncexp(-Edd/kT)
-> n = n0 exp(-Edd/kT) [at High Temp]
Therefore, at high Temperature, carrier concentration as function of deep donor ionization energy Edd can be written as either n = n0 exp(-Edd/2kT) or n = n0 exp(-Edd/kT).
Therefore, my significant query is which one is true for temp dependence of carrier concentration in case of dual dopant energy level model at high temp,
(i) n = n0 exp(-Edd/2kT)
or, (ii) n = n0 exp(-Edd/kT)
Kindly suggest the correct option. Your suggestion will be quite helpful for my research.
For your kind convenience I am attaching the reference book chapters & papers with necessary highlighted with yellow colour.
Dear Sen,
If you have two type of doping atoms, one is shallow and the other is deep.
Then you will have two activation energies one is large and and the other is small.
Normally the deep level will play only very small role in determining the the carrier concentration except at high temperature after depleting the shallow levels.
It is so then that the log conductivity versus 1000/T will have to distinct ranges.
One at relatively lower temperature where the shallow donor will be activated after which the material goes into saturation and it will resume the exponential growth again at the higher temperature range.
This what is observed for silicon with one shallow donor and thermal generation from the valence band which mimics the deep impurity levels.
To make the things plausible, let us assume silicon with one shallow donor .
At very low temperature say zero kelvin all donors are neutralized and no thermal generation from the valence band.
If we increase the temperature gradually the the shallow donors will be first start ionizing and electrons in the conduction band results which means an increase in the conductivity with temperature. This process continues till complete depletion of the shallow donors. In this case the deep donors or the valence band electrons as they need high temperature to be emitted from the valence band and so they start emitting electrons appreciably first at higher temperature. So, after depleting the shallow donor the conductivity remains almost constant until the deep donors begin to emit electrons to the conduction band and raise the conductivity further. So, we observe another exponential increase in the conductivity but with greater slope on Log sigma versus 1000/T.
I would like that you read my elementary book on electronic devices: Book Electronic Devices
I would like also that you see my paper:Article The dependence of diffusion length, lifetime and emitter Gum...
The paper is really of high quality but unfortunately remained with very low citations.
May it will find its luck after longtime.
Best wishes
Dear Sen,
If you have two type of doping atoms, one is shallow and the other is deep.
Then you will have two activation energies one is large and and the other is small.
Normally the deep level will play only very small role in determining the the carrier concentration except at high temperature after depleting the shallow levels.
It is so then that the log conductivity versus 1000/T will have to distinct ranges.
One at relatively lower temperature where the shallow donor will be activated after which the material goes into saturation and it will resume the exponential growth again at the higher temperature range.
This what is observed for silicon with one shallow donor and thermal generation from the valence band which mimics the deep impurity levels.
To make the things plausible, let us assume silicon with one shallow donor .
At very low temperature say zero kelvin all donors are neutralized and no thermal generation from the valence band.
If we increase the temperature gradually the the shallow donors will be first start ionizing and electrons in the conduction band results which means an increase in the conductivity with temperature. This process continues till complete depletion of the shallow donors. In this case the deep donors or the valence band electrons as they need high temperature to be emitted from the valence band and so they start emitting electrons appreciably first at higher temperature. So, after depleting the shallow donor the conductivity remains almost constant until the deep donors begin to emit electrons to the conduction band and raise the conductivity further. So, we observe another exponential increase in the conductivity but with greater slope on Log sigma versus 1000/T.
I would like that you read my elementary book on electronic devices: Book Electronic Devices
I would like also that you see my paper:Article The dependence of diffusion length, lifetime and emitter Gum...
The paper is really of high quality but unfortunately remained with very low citations.
May it will find its luck after longtime.
Best wishes
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