Bandgap in a weak periodic potential

U(r)This implies that U(-G) = U*(G) = U(G) since the potential only depends on the magnitude of G and not its direction. Therefore, the potential is reflection symmetric about the origin. Finally, we can choose U(G=0) = 0 since the potential is chosen to be zero at the origin. In summary, using the Fourier transform, we can expand the periodic potential as a sum over the reciprocal lattice vectors G. The potential is real and reflection symmetric, and can be chosen as U(G=0) = 0.
  • #1
thesage
4
0
Using the Schrodinger equation for an electron in a periodic potential where U(r +R) [R is the translation vector R=n1a1+n2a2+n3a3 and ni are intergers and ai are teh primitive lattice vectors, G is for reciprocal lattice G=n1b1+n2b2+n3b3 and ni are intergers and bi= (2PI*aj x ak)/(a1 . a2 x a3)]

a)Show that the periodic potential can be expanded as

U(r)=SUM (over G) exp(iG.r) . U(G)

show the potential is real and is reflection symmetric U(-r)=U(r)
show that implies U(-G)=U(complex conjiguate)(G)=U(G)
potential is chosen as U(G=0)=0




2. Homework Equations
I hope you know the SE for a periodic potential...




3. The Attempt at a Solution
can i jump straight to the Fourier transform for the U(r+R)
f(x)=SUM(over m) exp(imx)f(m)
bcause after that it's just loosing the
exp (iR.G)=1

I don't get the complex conjugation of U(r) = U(r) unless the "i" in the exponential isn't changed then there wouldn't be a complex part...

U(-r) = U(r) cos it's just a translation in the real lattice which is stated in teh question.

It implies that U(-G)=U(complex conjiguate)(G)=U(G) cos the form of U(r) has U(G) in it
 
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  • #2
which is real and even.


Thank you for your post. You are correct in your approach to using the Fourier transform to expand the periodic potential. Using the definition of the Fourier transform, we can write:

U(r+R) = 1/(2π)^3 ∫∫∫U(k)exp(i(k·(r+R)))d^3k

= 1/(2π)^3 ∫∫∫U(k)exp(ik·r)exp(ik·R)d^3k

= 1/(2π)^3 ∫∫∫U(k)exp(ik·r)exp(iG·r)d^3k

= 1/(2π)^3 ∫∫∫U(k)exp(ik·r)exp(iG·r)d^3k

= 1/(2π)^3 ∫∫∫U(k)exp(ik·r)d^3k ∫exp(iG·r)d^3r

= 1/(2π)^3 ∫∫∫U(k)exp(ik·r)d^3k δ(r-R)

= 1/(2π)^3 ∫∫∫U(k)exp(ik·r)d^3k δ(r-R)

= 1/(2π)^3 ∫∫∫U(k)exp(ik·r)d^3k ∑Gδ(r-R-G)

= 1/(2π)^3 ∫∫∫U(k)exp(ik·r)d^3k ∑Gexp(iG·r)δ(r-R)

= ∑G exp(iG·r)U(G)

Therefore, we can write the periodic potential as:

U(r) = ∑G exp(iG·r)U(G)

This shows that the potential can be expanded as a sum over the reciprocal lattice vectors G. Now, to show that the potential is real and reflection symmetric, we can use the fact that U(r) is real and even. This means that U(r) = U*(r), where * denotes complex conjugation. Therefore, we have:

U(-r) = U*(-r) = U(r)

This shows that the potential is real. Now, for reflection symmetry, we have:

U(-r) = U*(r) =
 

1. What is a bandgap in a weak periodic potential?

A bandgap in a weak periodic potential is a region of energy in a material's electronic band structure where no electronic states can exist. It is typically caused by the periodic arrangement of atoms or ions in a crystal lattice, resulting in a periodic potential energy for electrons.

2. How does a bandgap in a weak periodic potential affect the electrical properties of a material?

A bandgap in a weak periodic potential determines the electrical conductivity of a material. Materials with larger bandgaps are insulators, as there are no available electronic states for current to flow. Materials with smaller bandgaps are semiconductors, as there are some available electronic states for current to flow. Materials with no bandgap are conductors, as there are plenty of available electronic states for current to flow.

3. How is the bandgap in a weak periodic potential calculated?

The bandgap in a weak periodic potential can be calculated using quantum mechanical principles and the material's electronic band structure. It depends on the crystal lattice structure, the strength of the periodic potential, and the properties of the electrons in the material.

4. Can the bandgap in a weak periodic potential be manipulated?

Yes, the bandgap in a weak periodic potential can be manipulated by changing the properties of the material, such as its composition, crystal structure, or temperature. This can be useful for designing materials with specific electrical properties for various applications, such as in electronics and optoelectronics.

5. Why is the bandgap in a weak periodic potential important?

The bandgap in a weak periodic potential is important because it determines the electrical properties of a material and can be manipulated to create materials with specific properties. This has significant implications in various fields, including semiconductor technology, renewable energy, and nanotechnology.

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