A few simple questions about band structure and conduction

In summary: This is related to the concept of the Fermi level, which marks the boundary between filled and empty states in a band structure. Essentially, having an odd number of electrons per atom creates a situation where the highest energy state is only partially filled, rather than completely filled or completely empty.
  • #1
VortexLattice
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Hi! I'm reading about band structure and conduction in regular crystals and semiconductors, and I've hit a few confusing points my book doesn't explain well enough for me.

In one part, they're explaining conduction in a 1D crystal with lattice spacing [itex]a[/itex] (and therefore reciprocal lattice vectors [itex]G_n = 2n\pi/a[/itex]), and are showing the origin of band gaps. They give this diagram:

cWUFeEC.png


as the (extended/reduced/repeated) band structure. Then they say

nwPFFPs.png


(where the Bragg condition is [itex]n\lambda = 2a sin(\theta)[/itex] just like in a normal 3D crystal, but with [itex]\theta = \pi/2[/itex] for the 1D case.)

Here's where I'm confused: First of all, I don't know why "scattering admixes components with wave number [itex]k + G_n[/itex] for all [itex]n[/itex]." But I'll trust them on that, it seems reasonable. Secondly, they say that "in general the added components have different energy, so mixing is weak. It becomes strong only when the waves are of equal energy, which determines [itex]k[/itex]."

So then they set [itex]E(k) = E(k + G_n)[/itex] and solve for [itex]k[/itex] to find that it happens at [itex]k = G_n/2[/itex], which agrees with the Bragg condition. But, I'm confused. I thought, because of the nature of the Brillouin zone, [itex]k + G_n[/itex] has the same properties as [itex]k[/itex] for any [itex]k[/itex]. In fact, if you look at the repeated band structure, it really looks like if you choose any [itex]k[/itex] and hop over [itex]G_n[/itex], you're at the same height and thus have the same exact energy. So I'm a little confused why this condition isn't satisfied for any [itex]k[/itex].

My second question regards Bloch Oscillations. The book shows that electrons in an applied E field will oscillate rather than be accelerated uniformly. However, they say that electrons in a partly filled band will conduct, because they will be scattered to other states long before an oscillation is complete. So my question is: Would the electrons in a partly filled band not conduct if the crystal had no impurities and was very cold?

My last question is regarding this paragraph:

Wx8dPe1.png


jR6meag.png


I'm not sure why "an odd number of electrons per atom results in a half-filled top band", or really what they mean by "top band". I actually get the concept they're talking about, with the Fermi level, but not that sentence (/how it relates to the above band diagram I posted).

Any help would be much appreciated! Thanks!
 
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  • #2
For your first question, the idea is that when the wave numbers k and k + G_n are equal, the energy is also equal, so the scattering (which occurs when the wave numbers are different) is much stronger. This is because the electrons don't have to gain or lose energy to be scattered in this situation, so more electrons can be scattered. That's why this particular condition (k = G_n/2) is significant.For your second question, electrons in a partly filled band will still conduct even if the crystal has no impurities or is very cold. The reason is that there is still a net electric field from the applied field, which will cause the electrons to accelerate and give rise to Bloch oscillations. The difference is that, since there are no impurities, the electrons won't be scattered as they would in a normal crystal, so the oscillations will continue for some time before the electrons reach the boundaries of the system. For your last question, the "top band" refers to the highest energy band in the band diagram you posted. The reason an odd number of electrons per atom results in a half-filled top band is because the number of electrons is odd, and the total number of states in the band is even. So if all the lowest energy states are filled, then the highest energy state will be half-filled.
 

1. What is band structure?

Band structure refers to the arrangement of energy levels or bands within a solid material. It describes the distribution of energy states available to electrons in a material, which determines the material's electrical and optical properties.

2. How does band structure affect conduction?

Band structure plays a crucial role in determining a material's ability to conduct electricity. In materials with a large band gap, such as insulators, electrons are tightly bound to their atoms and cannot move freely, resulting in poor electrical conductivity. In contrast, materials with a small band gap or overlapping bands, such as metals, have more delocalized electrons and therefore exhibit good electrical conductivity.

3. What factors influence band structure?

The band structure of a material is primarily determined by its chemical composition and atomic arrangement. The number of valence electrons, the type of bonding (e.g. covalent, metallic, ionic), and the crystal structure all affect the energy levels available to electrons and, consequently, the material's band structure.

4. How does temperature affect band structure?

At higher temperatures, thermal energy causes atoms to vibrate more intensely, which can disrupt the regular arrangement of atoms in a solid. This can lead to changes in the band structure, such as increased band overlap, which can affect the material's conductivity. In some materials, temperature can also create additional energy levels within the band structure, leading to more complex electronic behavior.

5. Can band structure be manipulated?

Yes, band structure can be manipulated through various methods, such as doping (adding impurities to alter the number of electrons or holes), applying an external electric or magnetic field, or changing the material's temperature or pressure. These manipulations can change the band gap, band overlap, or the availability of energy states, which can have a significant impact on a material's electrical and optical properties.

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