Brillouin zones and the E-k relation

In summary, energy bands are formed in Brillouin zones, which are regions in reciprocal space that correspond to the periodicity of the crystal lattice in real space. In a 2D plane of atoms, the Brillouin zone edge in the k_y direction would be -pi/1.5a and pi/1.5a. The first energy gap occurs between reciprocal lattice vectors G_1 and G_4, with G_4 having a higher energy due to the propagation of the electron wave in the 2D plane. In a cubic structure, the first part is correct. For the second part, the notation of G1 and G4 is not commonly used for reciprocal lattice vectors.
  • #1
hhhmortal
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Hi, I'm having some trouble to fully understand how energy bands are formed in Brillouin zones.

Almongst a few of the questions I have are:

In a 2D plane of atoms, where a is the lattice constant in the x-direction and 1.5a in the y-direction. Would the brillouin zone edge in the k_y direction now simply be -pi/1.5a, pi/1.5a ?

Does the first energy gap occur between reciprocal lattice vector G_1 and G_4? And why does G_4 have a higher energy, I'm trying to visualise this in terms of the propagation of the electron wave in the 2D plane.

Thanks.
 
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  • #2
Well for the first part, if it's cubic, then yes that's right.

For the second part, I'm not sure about the terminology. What do you mean by G1 and G4? I am familiar with denoting reciprocal lattice vectors by Ghkl (or I guess two indices in your case).
 

1. What are Brillouin zones?

Brillouin zones are regions in the reciprocal lattice space that represent all possible wave vectors for electrons in a crystal lattice. They are used to describe the periodicity and symmetry of the crystal lattice structure.

2. How are Brillouin zones related to the E-k relation?

The E-k relation, also known as the energy-momentum relation, describes the relationship between the energy (E) and momentum (k) of an electron in a crystal lattice. The Brillouin zones are important in determining the allowed values of energy and momentum for electrons in a crystal lattice, as they represent the boundaries for these values.

3. How do Brillouin zones help us understand the behavior of electrons in a crystal lattice?

By mapping out the Brillouin zones, we can determine the allowed energy and momentum states for electrons in a crystal lattice. This information is crucial in understanding the electronic band structure and conductivity of materials, as well as other properties related to the behavior of electrons in a crystal lattice.

4. Can you explain the concept of the first Brillouin zone?

The first Brillouin zone is the smallest region in reciprocal lattice space that contains all possible wave vectors for electrons in a crystal lattice. It is bounded by the planes that are perpendicular to the major lattice planes and pass through the midpoint of the reciprocal lattice vectors.

5. How are Brillouin zones experimentally determined?

Brillouin zones can be experimentally determined through techniques such as electron diffraction or X-ray diffraction. These techniques allow for the measurement of the reciprocal lattice vectors, which can then be used to determine the boundaries of the Brillouin zones.

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