Are electrons in Valence band delocalized?

[rodsika]

Wikipedia says that "in band theory, energy bands are actually made up of many discrete energy levels which are too close together to resolve. Within a band the number of levels is of the order of the number of atoms in the crystal, so although electrons are actually restricted to these energies, they appear to be
able to take on a continuum of values".

Inquiry: Does it mean the electrons of a particular atom are still localized to that atom or are the electrons of that atom delocalized to the entire crystal.. meaning they can belong to any of the billions of atoms in that crystal??

 

[ZapperZ]

Wikipedia says that "in band theory, energy bands are actually made up of many discrete energy levels which are too close together to resolve. Within a band the number of levels is of the order of the number of atoms in the crystal, so although electrons are actually restricted to these energies, they appear to be
able to take on a continuum of values".

Inquiry: Does it mean the electrons of a particular atom are still localized to that atom or are the electrons of that atom delocalized to the entire crystal.. meaning they can belong to any of the billions of atoms in that crystal??

Let's look at 2 atoms when their orbitals overlap. You get things happening, including hybridization of these orbitals. In molecules, the bonding state no longer have localized electrons. These electrons are equally shared by all the atoms involved. Already these states do not look the same as isolated atoms.

Now take this even further and consider the tight-binding model. The overlap integral can involve the nearest neighbor, next-nearest neighbor, next-next-nearest neighbor, etc. By constructing such a system, you no longer have an electron localized at a particular atom.

Zz.

 

[rodsika]

Let's look at 2 atoms when their orbitals overlap. You get things happening, including hybridization of these orbitals. In molecules, the bonding state no longer have localized electrons. These electrons are equally shared by all the atoms involved. Already these states do not look the same as isolated atoms.

Now take this even further and consider the tight-binding model. The overlap integral can involve the nearest neighbor, next-nearest neighbor, next-next-nearest neighbor, etc. By constructing such a system, you no longer have an electron localized at a particular atom.

Zz.

In the Conduction band, the electrons are free to move or not localized. In the
Valence band, you are saying the electrons are not localized too. So how come no
electricity is produced in the valence band considering the electrons are not
localized too? What's the technical explanation why since both valence and
conduction band have delocalized electrons, only the conduction band can move
the electrons?
It looks like the electrons in the conduction band are being measured that's why
they precipitate into existence while the electons in the valence band are
hidden hence they are not physicalized as particles. Is this why when you put
electric potential difference, the valence band are not affected because the
electrons are not in physical form unlike in the conduction band? But why are
the electrons in the conduction band in particles state while the ones in
valence band in unparticles state?



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[DrDu]

Let's look at 2 atoms when their orbitals overlap. You get things happening, including hybridization of these orbitals. In molecules, the bonding state no longer have localized electrons. These electrons are equally shared by all the atoms involved. Already these states do not look the same as isolated atoms.

Now take this even further and consider the tight-binding model. The overlap integral can involve the nearest neighbor, next-nearest neighbor, next-next-nearest neighbor, etc. By constructing such a system, you no longer have an electron localized at a particular atom.

Zz.

Yes, but if the valence band is full (usual case), the corresponding set of orbitals can usually be transformed to a set of orbitals localized on two adjacent atoms. The same holds true in molecules. Hence I would not regard localization or delocalization of the electrons as an objective quality.

Therefore it is maybe more appropriate to ask whether the response to a specific local perturbation is localized or not, e.g. how far a hole generated in the valence band e.g. by X-rays propagates before it is filled some time later. In valence bands, hole mobility is typically low, i.e. on the order of the interatomic distance.

 

[ZapperZ]

In the Conduction band, the electrons are free to move or not localized. In the
Valence band, you are saying the electrons are not localized too. So how come no
electricity is produced in the valence band considering the electrons are not
localized too? What's the technical explanation why since both valence and
conduction band have delocalized electrons, only the conduction band can move
the electrons?
It looks like the electrons in the conduction band are being measured that's why
they precipitate into existence while the electons in the valence band are
hidden hence they are not physicalized as particles. Is this why when you put
electric potential difference, the valence band are not affected because the
electrons are not in physical form unlike in the conduction band? But why are
the electrons in the conduction band in particles state while the ones in
valence band in unparticles state?
* * *

The conduction band has empty states, allowing for mobility. The hopping integral in the tight-binding model will also tell you how these carriers can move. The valence band has no such empty states and so such movement are frustrated, but does it mean that the electrons are localized? Introduce a hole in the valence band and what do you get? If the carriers are localized at individual sites, such introduction would produce no difference.

Zz.

 

[rodsika]

Yes, but if the valence band is full (usual case), the corresponding set of orbitals can usually be transformed to a set of orbitals localized on two adjacent atoms. The same holds true in molecules. Hence I would not regard localization or delocalization of the electrons as an objective quality.

Therefore it is maybe more appropriate to ask whether the response to a specific local perturbation is localized or not, e.g. how far a hole generated in the valence band e.g. by X-rays propagates before it is filled some time later. In valence bands, hole mobility is typically low, i.e. on the order of the interatomic distance.

What do you mean you "would not regard localization or delocalization of the
electrons as an objective quality"?? Either they are localized or delocalized.
They can't be in a superposition of being localized and delocalized. Anyway. I
read the following in wikipedia which says it is delocalized
"It is a common misconception to refer to electrons in insulators as "bound"—as
if they were somehow attached to the nucleus and could not move. Electrons in
insulators are free to move. They are also delocalized, having no well-defined
position within the sample."
Or are there situations where the electrons are localized and delocalized?
Zapperz believes they are delocalized while you tend to believe they are
localized. Hope someone else can share their views.

 

[DrDu]

What do you mean you "would not regard localization or delocalization of the
electrons as an objective quality"?? Either they are localized or delocalized.
They can't be in a superposition of being localized and delocalized. Anyway. I
read the following in wikipedia which says it is delocalized
"It is a common misconception to refer to electrons in insulators as "bound"—as
if they were somehow attached to the nucleus and could not move. Electrons in
insulators are free to move. They are also delocalized, having no well-defined
position within the sample."
Or are there situations where the electrons are localized and delocalized?
Zapperz believes they are delocalized while you tend to believe they are
localized. Hope someone else can share their views.

I don't think that ZapperZ and I are really at variance at that point.
Electrons are quantum mechanical objects. You can describe even a highly localized electronic wavefunction in terms of completely delocalized ones. And in case of a many particle wavefunction, you often cannot tell apart which of the descriptions is to be preferred.

In chemistry one speaks of localized or natural bond orbitals, see:
http://en.wikipedia.org/wiki/Natural_bond_orbital
http://en.wikipedia.org/wiki/Localized_molecular_orbitals

The equivalent in solid state physics are Wannier functions:
http://en.wikipedia.org/wiki/Wannier_function

 

[rodsika]

I don't think that ZapperZ and I are really at variance at that point.
Electrons are quantum mechanical objects. You can describe even a highly localized electronic wavefunction in terms of completely delocalized ones. And in case of a many particle wavefunction, you often cannot tell apart which of the descriptions is to be preferred.

In chemistry one speaks of localized or natural bond orbitals, see:
http://en.wikipedia.org/wiki/Natural_bond_orbital
http://en.wikipedia.org/wiki/Localized_molecular_orbitals

The equivalent in solid state physics are Wannier functions:
http://en.wikipedia.org/wiki/Wannier_function

We must remember that orbitals are really probability waves. So are you saying
that in a molecular or crystalline object, the probability waves of all atoms
become entangled such that an electron in an atom in say the left most portion
of a square shape or cube crystalline object can simply "teleport" to the atom
of the right most portion?? This is what it means they are delocalized such that
the electrons of any part of the molecular can appear anywhere. But if this is
so. How come the charges and mass of an electron in an atom is still localized.
Note that if it were not localized, then the whole atoms and molecules would
just fall apart. With this context. How can the electrons be delocalized? If
they are delocalized, it implies that the charges and mass of the electrons are
delocalized too which is not possible or else the whole molecule would fall
apart.


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[DrDu]

Mabye the basic problem is that an electron on the left upper corner of a crystal is indistinguishable from an electron on the right lower corner. Therefore it makes not much sense to talk about an electron per se as being localized. However, it makes sense to ask how a localized excitation of an electronic system will propagate or whether for example the bonding in a crystal can be thought of to be approximately the sum of single bonds.
The different types of localized orbitals I mentioned are quite helpful to answer this kind of questions.

 

[rodsika]

Mabye the basic problem is that an electron on the left upper corner of a crystal is indistinguishable from an electron on the right lower corner. Therefore it makes not much sense to talk about an electron per se as being localized. However, it makes sense to ask how a localized excitation of an electronic system will propagate or whether for example the bonding in a crystal can be thought of to be approximately the sum of single bonds.
The different types of localized orbitals I mentioned are quite helpful to answer this kind of questions.

Hi, ok I got your point. But I can't understand this previous statement of yours
when you stated: "Yes, but if the valence band is full (usual case), the
corresponding set of orbitals can usually be transformed to a set of orbitals
localized on two adjacent atoms." What "corresponding set of orbitals" are you
talking about? And what do you mean by "can usually be transformed to a set of
orbitals localized on two adjacent atoms"? What are you describing here?? Pls.
rephrase it, because your vague statement is what got me so confused in this
thread. Thanks.


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[DrDu]

I gave you three wikipedia links on that topic. However, these may be to technical for you.

Any chemist will describe a molecular crystal like solid H2 in terms of localized orbitals on the hydrogen molecules or quartz in terms of localized bonds between Si and O and not in terms of delocalized valence band states, which are the default choice of the solid state physicist.
(In a molecular crystal like solid benzene on the other hand, it is not possible to localize all orbitals between two atoms. That's why I used the phrase "usually". )
The band picture and the picture of localized orbitals are mathematically equivalent if the electrons are treated on Hartree Fock level, i.e. in an effective one electron picture.

 

[rodsika]

I gave you three wikipedia links on that topic. However, these may be to technical for you.

Any chemist will describe a molecular crystal like solid H2 in terms of localized orbitals on the hydrogen molecules or quartz in terms of localized bonds between Si and O and not in terms of delocalized valence band states, which are the default choice of the solid state physicist.
(In a molecular crystal like solid benzene on the other hand, it is not possible to localize all orbitals between two atoms. That's why I used the phrase "usually". )
The band picture and the picture of localized orbitals are mathematically equivalent if the electrons are treated on Hartree Fock level, i.e. in an effective one electron picture.

Ok. Understood. Say. What is the difference between the valence band in metal
versus crystal? I know that in metal the band gap is very small or nonexistent
which can make the electrons jump to conduction band while the crystal has huge
band gap making it an insulator. But I'm asking specifically what is the
difference in energy band quantity and quality in the *valence band* only
between metals and crystals, like do they have similar wideness and energy
range?

 

[DrDu]

The simplest metals only have a conduction band and no valence band. In those metals which have a valence band, its properties are not fundamentally different from valence bands in insulators.

 

[rodsika]

The simplest metals only have a conduction band and no valence band. In those metals which have a valence band, its properties are not fundamentally different from valence bands in insulators.

Interesting. I've been searching at web sites but can't seem to find the
information of how to tell what metals have only conduction band and no valence
band. So you mean simple metals like copper doesn't have valence band. I wonder
what is the rule to determine what metals would have valence band and those that
don't.

 

[DrDu]

Copper is not necessarily simple. I thought more of e.g. Alkali metals.

 

[rodsika]

Copper is not necessarily simple. I thought more of e.g. Alkali metals.

I see.
Anyway. The concept of energy band only occurs in crystalline matter which
includes metals. So wood and plastics don't have any energy band? Meaning the
orbitals are isolated in each atom? But since* wood or plastic molecules are
connected together to make a continuous object such as a plastic cup or wooden
chair, then how could they be connected without forming energy bands?**