Why some solids are electrical conductors

[Kawakaze]

Homework Statement

Describe what influence the periodic structure of a crystalline solid has on the energy distribution of the free electrons, and hence explain, in general terms, why some solids are electrical conductors and some are insulators.

Homework Equations

The Attempt at a Solution

Im assuming the question can be partly answered with band theory. What I don't understand is what bearing the crystalline structure has. Please note this is for an assessment, so no direct answer please. Band theory refers to the sub-shells and the energy gaps between them, can I have a hint on how the crystalline structure may affect this? Is it something to do with bonding, ionic or covelant and how can this even factor in something like a pure metal sample? Thanks =)

 

[Dickfore]

Bloch's Theorem.

 

[Kawakaze]

Hi Dick, thanks for the answer, unfortunately that isn't in my course material at all. Maybe a little higher than the level I am doing. I did have a read of it, and it does mention the same two models i am aware of, namely the nearly free electron model and the tightly bound electron model. I still cannot see how this all fits together though. Maybe another hint please? =)

 

[Dickfore]

What course are you taking and what is the course material you are using?

Have you ever met some of these terms:

1. Direct Lattice
2. Reciprocal Lattice
3. First Brillouin Zone
4. Born von Karman Periodic Boundary Conditions
5. Density of States

Essentially, the free-electron (+hole) model is a fairly good approximation when analyzing the electric properties of solids. But, instead of using plane waves as the stationary states for the electron, the corresponding basis is spanned by the above-mentioned Bloch states. Two main effects arise from this modification:

1. The energy levels are split into bands of allowed and forbidden intervals;
2. The electrons (and holes) acquire an effective mass (which might be a second - rank tensor quantity) which has a profound effect on the density of states

 

[rdl]

The periodic structure of a crystalline solid plays a crucial role in determining its electrical conductivity. In a crystalline solid, the atoms are arranged in a regular, repeating pattern, which creates a well-defined energy band structure. This band structure consists of allowed and forbidden energy levels for the electrons in the solid.

In general terms, the electrons in a solid can be classified into two types: bound electrons and free electrons. Bound electrons are tightly held by the atoms and cannot move freely in the solid, while free electrons have enough energy to move throughout the solid. In a perfect crystal, the valence electrons (outermost electrons) are in the highest energy level of the atom, called the valence band.

The periodic structure of a crystalline solid influences the energy distribution of the free electrons. In a pure metal, the valence electrons are not bound to a specific atom and are able to move freely throughout the solid. This is because the atoms are arranged in a regular pattern, which creates a continuous energy band that allows for the free movement of electrons. This results in high electrical conductivity.

On the other hand, in an insulator, such as a ceramic or a non-crystalline solid, the atoms are not arranged in a regular pattern and the energy band structure is disrupted. This leads to a large energy gap between the valence band and the conduction band, making it difficult for electrons to move freely and thus resulting in low electrical conductivity.

In summary, the periodic structure of a crystalline solid plays a crucial role in determining its electrical conductivity by influencing the energy distribution of the free electrons. This is why some solids, like pure metals, are good electrical conductors, while others, like insulators, are poor conductors.