What is the Threshold for Interband Optical Transitions in a 2D Square Lattice?

In summary, for a two dimensional solid with two electrons per unit cell, a Bravais lattice with primitive vectors ##\vec a = \ell \hat x## and ##\vec b = \ell \hat y##, and a weak crystal potential, the solid behaves like a free electron metal. The first Brillouin zone and Fermi contour can be sketched in a reciprocal space diagram. The energy band diagram shows the Fermi energy cutting two bands, allowing for partial filling and conduction. Estimating the threshold for interband optical transitions in terms of ##\epsilon_f##, the Fermi energy, requires considering the promotion of an electron from one band to another with the help of an external means in the eV
  • #1
CAF123
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Homework Statement


A two dimensional solid has two electrons per unit cell and has a Bravais lattice with primitive vectors ##\vec a = \ell \hat x## and ##\vec b = \ell \hat y##. The crystal potential is weak and the solid behaves like a free electron metal.

a)In a reciprocal space diagram, sketch the first Brillouin zone and Fermi contour.
b) Draw an energy band diagram for this solid
c) Estimate the threshold for interband optical transitions in terms of ##\epsilon_f##, the Fermi energy.

Homework Equations


2 electrons per unit cell imply first B.Z fully occupied.

3. The Attempt at a Solution

a) and b) are done I think, I am just wondering really how to begin with c). By interband transition, I think it means we are considering a transition or promotion of an electron in one band to another. It does so by some external means giving it enough energy to overcome the ##\epsilon_f## energy gap. However such a means is not specified.

The diagram in b, given that the crystal potential is weak, shows the fermi energy cutting two bands, so each band is partially filled and conduction can take place. I am just now sure how to begin. If the crystal potential was stronger, the band gap opens and the solid tends toward an insulator.

Many thanks for any help!
 
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  • #2
Hmm, can you show us a picture of your answer to b)?

As to the means, it says "optical". Implying energies in the eV range and negligible momentum transfer.
 

Related to What is the Threshold for Interband Optical Transitions in a 2D Square Lattice?

1. What is a 2D simple square solid?

A 2D simple square solid is a two-dimensional shape that has four equal sides and four right angles. It is a flat shape that is often represented by a square or a rectangle.

2. How many dimensions does a 2D simple square solid have?

A 2D simple square solid has two dimensions - length and width. It does not have a third dimension, such as height or depth.

3. What are the properties of a 2D simple square solid?

The properties of a 2D simple square solid include having four sides of equal length, four right angles, and opposite sides that are parallel and equal in length.

4. How is a 2D simple square solid different from a 3D square solid?

A 2D simple square solid is a flat shape with only two dimensions, while a 3D square solid has three dimensions and is a solid object. A 3D square solid has length, width, and height, while a 2D simple square solid only has length and width.

5. What are some real-life examples of a 2D simple square solid?

Some real-life examples of a 2D simple square solid include a book, a piece of paper, a tile, and a checkerboard. These objects have a flat shape and can be represented by a square or a rectangle.

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