I was getting mixed between the description of 'electron traps' (for example, donor like traps) and that of 'impurity states' (let’s assume donors). Apparently, in terms of the band diagram, both are treated as states within the band gap (they may lie within the bands also, I am not sure, but definitely they can be inside the band-gap). Both are characterized by their energy, density, degeneracy, etc. Yet there seems to be a difference between an impurity state at some energy level (with respect to a reference, let’s say, the valance band), and a trap state, situated at the same energy level measured with respect to the same reference. I understand that their functions are different, but can that difference be explained using the energy of traps/impurity states and the (Fermi Dirac) carrier statistics? The ionization of impurity state is determined by the energetics (the relative energy difference from Fermi level, the density, the degeneracy). The ionisation of traps is, however, involving some kinematic factors like the trap life time.
I also read in an article that the population of electrons in trap states are described by a non-equilibrium but steady-state process. What I did not understand was, why so.
Any thoughts please?
No, it is not true in general. Traps are just impurity levels deep inside the fundamental gap of the host material. Of course, a particular sample may have been prepared in such a way that impurities are not in equilibrium in their host system. I think you must be referring to the following paper by Simmons and Taylor:
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.4.502
If so, considering traps under non-equilibrium steady-state conditions is an issue different from trap states being "described by a non-equilibrium but steady-state process". Consider the fact that a current-carrying system may be in a steady state, but it is not in an equilibrium state.
I think that it is not true in general. Both shallow (EMT) impurities and deeper states (so cold TRAPS) can be ionized (or neutral) in thermal equilibrium conditions or in a kinetic related processq such as optical absorption or emission. The latter is sometimes treated in the so called "quasiequilibrium approach" via Quazi-Fermi level, but usually this approach does not work properly. The better approach is kinetic-one and it was applied first for the traps due to the fact, that the ionization processes are realized at higher (really - very high) temperatures. The correct treatment of light emission processes of free electrons developed from the shallow impurities (so cold FERB) in GaAs, InP, GaN and InN is possible only by cinetic approach, as shown recently.
Thanks Behnam and Arnaudov,
Yes, I was referring to the article by Simmons and Taylor, and the picture in my mind was exactly what Behnam said, of a current carrying system, which is in steady state, but not in equilibrium. I am using the kinematic (rate limited) approach for simulating effect of traps in GaN/AlGaN.
As the other extreme, I was wondering, if any of you have a reference where traps are in equilibrium with the rest of the system, and the population of electrons/holes in them is described by the F-D statistics? The kinematic approach that I am using, has an expression for trap occupancy that approaches the F-D occupancy at high values of the capture cross section and I was trying to get the physical picture of that. Thanks a lot, to both of you.
You are welcome S. Bhattacharyya. In general, if the sample is prepared well and no current is transported through it, traps are in equilibrium. In such case, given the energy levels of the system, the number of charged particles and the temperature, all that needs to be calculated is the chemical potential, and one has the thermal distribution of the particles in the system, including the distribution of the particles inside the traps. No special literature is required for this. The situation is different when current is carried through the sample, in which case one needs to calculate the distribution of the charged carriers (often the steady-state distribution function, but sometimes also the transient distribution function). For this, one can use some transport equation, such as the (linearised) Boltzmann equation and the equation of motion for the Wigner distribution function (the latter is relevant to transient distribution function -- the function before the steady-state sets in). For the former, you may wish to consult the following paper, which is standard work in the field, by Gerald Mahan (Mahan's book, Many-Particle Physics, has also good sections on transport theory):
http://www.sciencedirect.com/science/article/pii/0370157387900044
See also this (the entire book is dedicated to transport theory):
http://link.springer.com/chapter/10.1007%2F978-1-4899-2359-2_6
For the latter, you may consult the following review article by William Frensley:
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.62.745
In general, using the Keldysh formalism, one can relatively straightforwardly deduce transport equations. Within this formalism, one can apply controlled approximations, which is necessary for making solution of the relevant equations practically feasible. See for instance the following review article by Rammer and Smith:
http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.58.323
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