Understanding Free Electron Kinetic Energy on a Square Lattice

In summary, the question is asking to show that the kinetic energy of a free electron is 2 times higher at the corner of the first zone compared to the midpoint of a side face in a simple square lattice. The solution involves using a tight binding model with nearest neighbor hopping and considering the electron's position in k-space. The square lattice specifies the shape of the Brillouin zone and determines the energy spectrum.
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Homework Statement


Show for a simple square lattice that the kinetic energy of a free electron is higher at the corner of the first zone than at the midpoint a side face by a factor of 2.


Homework Equations


Simple geometry.


The Attempt at a Solution


I think I know how to solve, but I am not completely sure I understand the question. The position of the electron refers to the position of its k-vector in k-space right? Well this is not real space, and I can't really see how this has anything to do with the lattice being square in real space.
 
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  • #2
I suspect the hidden assumption here is that you should use a tight binding model with nearest neighbor hopping on the lattice given. So in your case the square lattice is specifying the shape of the Brillouin zone and telling you the energy spectrum.

Does this help?
 

Related to Understanding Free Electron Kinetic Energy on a Square Lattice

What is a free electron square lattice?

A free electron square lattice is a model used in condensed matter physics to describe the behavior of electrons moving in a periodic potential. It consists of a square array of atoms or molecules, with each lattice point representing a potential energy well that can trap an electron.

How do electrons behave in a free electron square lattice?

In a free electron square lattice, the electrons behave as if they are free particles, with no interactions between them. They can move through the lattice without any resistance, similar to how electrons move through a vacuum.

What is the band structure of a free electron square lattice?

The band structure of a free electron square lattice is characterized by a series of energy bands and gaps. The lowest energy band, known as the valence band, is completely filled with electrons. The next higher energy band, known as the conduction band, is empty or partially filled. The size of the band gap between these two bands determines the material's electrical conductivity.

What is the importance of a free electron square lattice in materials science?

The free electron square lattice model is important in materials science because it helps explain the properties of different materials, such as their electrical conductivity, thermal conductivity, and optical properties. It also provides a basis for understanding the behavior of electrons in metals, semiconductors, and insulators.

What are some limitations of the free electron square lattice model?

The free electron square lattice model is a simplified version of how electrons behave in real materials, and it has some limitations. It does not take into account the effects of electron-electron interactions, which can significantly affect the electronic properties of a material. It also does not consider the crystal structure and atomic arrangements of a material, which can affect the electron behavior. Additionally, it cannot explain the behavior of materials at very low temperatures or under extreme conditions.

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