Density of states from 3D to 2D

In summary, the conversation discusses how to calculate density of states for both the 3D and 2D cases, with a focus on transitioning from the 3D case to the 2D case. This is demonstrated by manipulating the formula for energy from an infinite potential well and discussing the effects of changing the dimensions. It is noted that if one dimension is extremely small, the system is reduced to 2D with the only possible value for n_x being 0. Alternatively, making the dimension very big would also result in a 2D system.
  • #1
Matej Kurtulik
9
0
Hi,
I know how to calculate density of states for both cases, but it is not clearly to me how I can go from 3D case to 2D. I have energy from infinite potential well for 3D
$$E=\frac{\hbar \pi^2}{2m}(\frac{n_x^2}{l_x}+\frac{n_y^2}{l_y}+\frac{n_z^2}{l_z})$$
let make one dimension very small
$$l_x<<1$$
I should come to 2D conclusion, but I don't see any difference.

Thanks
 
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  • #2
If ##l_x## is extremely small, then the only possible value of ##n_x## is 0, as the energy for excitation would be too big. The system is then reduced to 2D.
 
  • #3
You could try making it very big instead.
 

Related to Density of states from 3D to 2D

1. What is the difference between 3D and 2D density of states?

The density of states describes the number of quantum states per unit energy in a given system. In 3D, the density of states is a function of energy, while in 2D, it is a function of both energy and momentum. This means that the density of states in 2D is highly dependent on the direction of the momentum, while in 3D it is not.

2. How is the density of states affected when transitioning from 3D to 2D?

When transitioning from 3D to 2D, the density of states can change significantly. In 3D, the density of states is typically constant, while in 2D it can vary greatly with energy and momentum. This is due to the confinement of particles in 2D systems, which leads to a discrete energy spectrum and changes in the momentum distribution.

3. What is the importance of studying the density of states in 3D to 2D transitions?

The density of states is an important concept in understanding the electronic properties of materials. In 3D to 2D transitions, changes in the density of states can greatly affect the electronic behavior of materials, such as their conductivity and optical properties. Understanding these changes is crucial for the development of new materials for various applications.

4. How can the density of states be calculated in 3D and 2D systems?

The density of states can be calculated using mathematical equations and models based on the energy and momentum distribution of particles in a given system. In 3D, the density of states can be calculated using the free electron model or quantum mechanical models. In 2D, it can be calculated using the 2D electron gas model or other appropriate models for specific systems.

5. What are some real-world applications of understanding the density of states in 3D to 2D transitions?

Understanding the density of states in 3D to 2D transitions has numerous real-world applications. For example, it is crucial in the development of advanced electronic devices, such as transistors and solar cells, as well as in the study of thin films and nanostructures. It also plays a significant role in the fields of materials science, condensed matter physics, and nanotechnology.

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