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I have some confusion about the concept of some electronic bands and energy levels. Beyond valance band,

For an atom, can having at least one electron in the conduction band mean that the atom has been ionised?

Can having an electron in the donor energy level or acceptor energy level mean that the atom has been ionised?

Is free electron same as an electron in the conduction band or an electron in the acceptor or donor band?

Is an atom having a free electron an ionised atom?

I am super-super-confused...

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2 Answers 2

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The band theory of solids has developed to describe the collective behavior of atoms and molecules when in the solid form.

bands

The confusion goes away if one realizes that different quantum mechanical models have developed in order to describe different problems.

For an atom, can having at least one electron in the conduction band mean that the atom has been ionised?

In order for an electron to be considered in the conduction band, one needs a solid. Individual atoms/molecules sit at the lattice sites of the solid. The model is that the conduction band has electrons that are bound over the whole lattice. The valence band electrons are tied on the points of the lattice where the nucleus also sits (depending on the material, whether insulator or conductor).

In the sense of the band theory model, the whole lattice is still neutral. Just the conduction band electrons have definition over the whole lattice ( a quantum mechanical probability to be anywhere in the lattice). It has little meaning to talk about an ionized atom, because its "own electron might be away but another in the conduction band is near", (as a hand-waving explanation).

Can having an electron in the donor energy level or acceptor energy level mean that the atom has been ionised?

If for individual atoms, yes, but if the atom is part of a lattice see above.

Is free electron same as an electron in the conduction band or an electron in the acceptor or donor band?

No a free electron means there is no potential that is binding it.

Is an atom having a free electron an ionized atom?

If the atom is not in a lattice and it attracts a floating electron, yes it will be a negatively ionize atom. If in a lattice,see above

Maybe if you read the whole link your confusion will go away.

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  • $\begingroup$ So, a free electron is sourced from a positively ionized atom? And a free electron does not have any connection to any energy and band levels since there is no electrical potential binding it?? $\endgroup$
    – PhysicsSolvesAll
    Commented Jun 14, 2021 at 18:41
  • $\begingroup$ yes, that's what free means. An electron in a solid is never "free", it is in an energy level of the lattice ,instead of just an atom. $\endgroup$
    – anna v
    Commented Jun 14, 2021 at 18:45
  • $\begingroup$ Is it correct to say: "For a single atom any energy above valance energy level is free electron, but for a solid lattice, anything above conduction level is free electron aka positively ionized(loss of electron by freeing it with absorption of required energy) atom then? @anna v $\endgroup$
    – PhysicsSolvesAll
    Commented Jun 14, 2021 at 19:17
  • $\begingroup$ No. You are confusing in the statement the two models, the standard atomic model with the electrons energy levels according to the potential between the lectrons and the nucleus, and the term "valence" has no meaning in this model, the term "free electron" has. and the band theory of solids with a collective potential for a large number of atoms at lattice points tied by chemical bonds when looked at individually, but fitting collectively the band theory QM model. Here belongs the "valence" and not the "fee electron". $\endgroup$
    – anna v
    Commented Jun 15, 2021 at 4:07
  • $\begingroup$ Hand waving you can call the electrons in the conduction band "free" to wander the whole lattice, and in conductors even the valence quarks can be described by handwaving as able to move through the lattice (see link), they are all not free to various degrees but at lattice energy levels. $\endgroup$
    – anna v
    Commented Jun 15, 2021 at 4:07
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Ionization requires an atom/molecule as a whole, should have a net positive charge i.e. an electron has to be completely removed from the neutral atom/molecule. Strictly speaking, addition of electrons to a neutral atom is also a kind of ionization (and hence the name AN'ION') and so the most general definition nessesitates the requirement of a NET CHARGE on the atom/molecule.

In all the occasions you have mentioned the atom/molecule is neutral, hence not ionized. To be more precise, ionization is a extreme Excitation when the excitation energy equal (or greater than) Ionization Energy of that state.

Now all the situations you have mentioned are either ground state or excited states but not Ionized state.

Usage of the term "Free Electrons" -

  1. Free electrons/particles occur when there is no BOUND POTENTIAL in the Hamiltonian. It is commonly characterized by a infinitely whole range energy levels with no forbidden states poking around.
  2. Misnomer in Solid State Physics - Any solution of Schrodinger Equation that represent the Plane Wave solution is commonly called as Free Electron. It is important to note that strictly speaking these are not "free Electron" although they don't explicitly have a Potential in the Hamiltonian, the bound potential is implicitly taken into consideration in the Effective Mass. This approach simplifies the calculation, but doesn't change the physics. Hence, electrons in the various bands of the Lattice are in fact bound electrons, although some of them may be referred to as free electrons! Another effect of these bound electrons is that you have to provide some quanta of energy to release it from the lattice.
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  • $\begingroup$ Is free electron and electron in conduction band same thing? $\endgroup$
    – PhysicsSolvesAll
    Commented Jun 14, 2021 at 17:57
  • $\begingroup$ so, both "an electron in the conduction band" and "free electron" sources from an ionised(anion) atom? $\endgroup$
    – PhysicsSolvesAll
    Commented Jun 14, 2021 at 18:02
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    $\begingroup$ ohh... not exactly. Current (in out context) is a phenomena that occur due to drift of electrons and not sourcing of electrons. Batteries doesn't source electrons they simply provide potential so that electrons already present in the wire/conductor can drift across. So, in our context as soon as one electron leaves an a nuclei other electron behind it takes its place, essentially keeping the conductor (and hence the entities that make up the conductor) neutral! $\endgroup$
    – TheImperfectCrazy
    Commented Jun 14, 2021 at 18:15
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    $\begingroup$ No its wrong to regard an electron in conduction band as a free electron, simply because this is a bound system and so electrons are "bound" and no free to go outside the metal. The particular phrase is a misnomer that got stick in literature probably because of the fame of Free Electron Model. $\endgroup$
    – TheImperfectCrazy
    Commented Jun 14, 2021 at 18:33
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    $\begingroup$ Ahh... Mobile electrons are a much better word to use. It would to correct to say that "electrons in conduction band are mobile electron." All electrons above valence band need not be mobile electrons. I cant think of any example right now, but in general you can have certain bands above conduction band that need not be conduction bands. This commonly happens when you have a very large energy gap above the conduction band especially due to topological effects. $\endgroup$
    – TheImperfectCrazy
    Commented Jun 15, 2021 at 7:41

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