When looking at a P type semiconductor band diagram, the acceptor level sits a little higher than the edge of the valence band, say Boron in Silicon. i have learned that the acceptor level represents the energy needed for an adjacent silicon valence electron to jump to the incomplete covalent bond(frustrated silicon atom with 3 valence electrons) and thus forming a stronger covalent bond, since that electron now belongs to the no longer frustrated silicon atom and is not shared with any other adjacent atoms. If that is the case, that this incomplete covalent bond attracts a nearby electron, then the energy state of this new electron when stolen from the atom should be more convenient(nature tries to keep the energy of particles to the lowest levels), so the acceptor level should be BELOW the valence band edge.. ? I also read the acceptor level is something mathematically derived using the Hydrogen atom model analogy with modified permitivity and effective mass and the result puts the ionisation energy to that level. I am quite confused here :)
1 Answers
You've arrived at the seeming contradiction because you think in the direction opposite to what is productive from the pragmatic point of view, I mean, what you are doing when computing the eigenstate shift caused by a lattice imperfection due to an acceptor atom. Think the other way round: rather than a "stolen" electron, you have a positive hole localised "outside" the boron atom and weakly held by the negative charge of the acceptor atom enriched with an excessive electron, and it is this quasi particle (hole) that is exchanged with adjacent bonds.
You can read a tolerable explanation of this mechanism in Chapter 9, page 6 of https://pages.hep.wisc.edu/~prepost/623/transistor/pnlecturenotes.pdf with a hole mass and the silicon permittivity participating in the binding energy formula. The "Hydrogen atom model analogy" is acceptable because a potential well that gives the binding energy of 0.1eV is really shallow. I admit the last statement smells a logical fallacy of begging the question; the strict proof is left to OP.
UPDATE in response to comments:
Let us apply your consideration of covalent bond formation with participation of the acceptor atom to computation of wavefunctions and energy levels. Suppose, in computation we decided to directly address the "stolen" electron, participating in covalent bonds. But the acceptor atom is surrounded by multiple silicon atoms, so which one donates the electron? Even with a classical physics consideration, we have at least four nearest candidates. You consider one "frustrated silicon atom which has 3 covalent bonds", but there are a good few "frustrated silicon atoms" that share their loss, totaling one electron minus, within their group. Notice that any silicon atom, participating in the formation of covalent bonds with the acceptor atom, still forms all four covalent bonds, despite one electron missing. More precisely, there are no individual "frustrated" silicon atoms, but all four covalent bonds, formed with the acceptor atom, are "frustrated", because the acceptor atom has only three valence electrons, and the "collective" wavefunction of these four covalent bonds is deficient of one electron.
Therefore, you cannot trace just one individual electron when solving the problem of the acceptor atom in the silicon lattice. There are many electrons with essentially overlapping individual wavefunctions. This category of physical problems is called "many body problems", the "collective" wavefunction being a many body wavefunction. These problems are hard to solve brute-force.
Fortunately, the problem of lattice imperfections is amenable to solution with a Schroedinger equation for individual particles, where the collective motion of participating covalent electrons is modeled as an equation of motion for a quasi particle. Strictly speaking, the contribution of a mobile carrier donated by the donor atom is also computed as a quasi particle called "electron quasiparticle", with an effective mass different from the electron mass, slightly for Si and not-so-slightly for the other semiconductors. The mobile carrier donated by the acceptor atom is called a hole.
One more reference with excellent visuals: http://ee.sc.edu/personal/faculty/simin/ELCT566/03%20Semiconductors%20II-Doped,%20transport.pdf .
As an exercise, re-draw the slide "Concentration-temperature dependence in doped semiconductors" (page 6) to illustrate the case of p-Si.
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thanks for your reply once again. i read the linked pdf, however that still does not answer my question as to what the acceptor level represents. – pit fermi Dec 20 '20 at 00:29
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for the donor level, we say it is the minimal energy needed for the electron to be elevated to the conduction band to move free in the silicon crystal(since conduction band for silicon means free electrons in the silicon lattice). now looking at an acceptor and the frustrated silicon atom which has 3 covalent bonds. at room temperature, the kT is enough for an adjacent Silicon electron to jump in that crystal imperfection. does the electron belong to the silicon now, or is it part of the acceptor atom? if the energy needed for a valence electron to jump in that electron absence/crystal – pit fermi Dec 20 '20 at 00:38
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imperfection, it is bound to the silicon and cannot go away anymore, since this electron belongs to silicon now and is not shared. it is still confusing – pit fermi Dec 20 '20 at 00:40
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In a precise analogy with your consideration of donor-doped silicon, the minimal energy required for the bound hole to be elevated to the conduction band to move free in the acceptor-doped silicon lattice -- since conduction band for acceptor-doped silicon means free holes in the acceptor-doped silicon lattice -- is what we call the acceptor level, and the acceptor level is the minimal energy defined in this way and nothing else. For more detailed explanation, see the edit to my answer. – V.V.T Dec 20 '20 at 07:25