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From http://education.jlab.org

In a metal, the atoms are arranged in a crystal-like configuration.

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Now, in a metal, the valence band is relatively close to the conduction band - that is, very little energy is necessary to cause electrons to jump from their valence state into the conduction band. In fact, we think of metals as having a large population of free electrons in the conduction band all the time. So the application of electric potential will cause them to move - a current flow. So, metals generally have a relatively low (though not zero) resistance. In a material such as glass, there is a large energy gap between the valence and conduction band. This means there are very few free electrons available for current flow, and it takes a large input of energy to raise any electrons into the conduction band.

Why is it related to crystal-like structure? Glass does not have crystal structure therefore distance between valence and conduction bands is high?

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The basic idea is that if you have a regular periodic lattice then the individual electron wave functions of the atoms can combine together to make electronic wave functions which extend through the entire lattice. That doesn't happen if the material does not have a regular periodic lattice. If the material doesn't have a regular periodic lattice, the individual electron wave functions can still combine with each other, but the resultant wave functions don't extend through the entire lattice of the material.

So if you have a periodic lattice then the electronic wave functions extend through the entire solid, but even so you can still have an insulator, not an electrical conductor. Look at a grain of table salt. It's a nice little cube because its almost a perfect single crystal with a regular cubic periodic lattice. The electronic wave functions in it extend all across the crystal. But yet it doesn't conduct electricity. Why? Because there's another factor to be considered: For an electron to absorb energy and move from one side of the crystal to the other, there has to be an empty available energy level for it to occupy. For table salt, there are no nearby empty energy levels, so salt remains an insulator despite the fact that its electronic wave functions extend thoughout the entire crystal. In other words, there is an "energy gap" as stated in the quote you presented. For crystalline metals, there are available excited energy states, so they can conduct electricity quite well.

A periodic lattice is not required for electrical conduction and indeed there are metallic glasses and metallic liquids. However, since the electrons cannot "sail" through a nice periodic lattice on wave functions which extend though the entire length of the material, the electrical conductivities of metallic glasses and metallic liquids tends to be significantly lower than those of crystalline metals.

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I think almost explanation for solid state physics assumes that the solid has periodicity(crystal structure) for simplicity. If we assume the periodicity condition(crystal structure), We can give a system a periodic potential and easily solve Schrodinger equation. Also, Materials which show good physical properties usually have crystal structure in real life.

Although the system has no crystal structure(no periodic potential, Amorphous materials), We can solve Schrodinger equation, get energy state of electrons. However, It is extremely difficult to know a potential form of system for having no periodic structure.

why do you think glass have no crystal structure? Materials have various structure under many condition(temperature, pressure, ...). ex) Even for quartz(SiO2), α-quartz has rhombohedral (trigonal crystal geometry. β-quartz has hexagonal crystal geometry.

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  • $\begingroup$ Exactly my point - glass and many other materials have crystal structure. When explaining conductance I see mention of crystal structure as explanation which's not helpful because we envision electrical flow like that. Also semiconductors are formed by doping. Water with impurities is a conductor seemingly without crystal structure. This explanation probably is so popular because it's easy to make mental image. $\endgroup$ – pusheax Sep 26 '15 at 17:11
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One has to remember that necessary (crystal structure) is not also sufficient ( for having conductivity) . For conductivity the specific atoms have to have a small energy interval to the conduction band, the band shared with the whole crystal lattice.

energybands

It takes very little energy to raise an electron from a valence state to the conduction band of the crystal , in conductors. Energy which is provided by the voltage imposed.

So the crystal lattice is necessary in order to model the bands, but conduction happens when the conduction band has a small energy difference from the valence band.

This is a very successful model in describing the behavior of solids to electric fields.

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