Band diagram in real space vs reciprocal space

In summary, the relationship between the band diagram in k-space and the diagram in real space is related to the variation of carriers concentration with the position.
  • #1
itler
8
0
Hi,

can anybody rigurously explain the relationship between the band diagram in k-space (I think I understand this one) and the diagram in real space (as is often used to explain the p-n-junction).
 
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  • #2
Diagram in real space shows the dependence of band edge E[tex]_{C}[/tex] (r) or E[tex]_{V}[/tex] (r) to the position r inside the system.
This dependence is related to variation of carriers concentration with the position n(r), p(r); typical of inhomogeneous doped material (Ex: pn junction).
In this system the total current density is the sum of two term: 1) drift contribute (proportional to external electric field) and 2) diffusional contribute (proportional to the gradient of carriers concentration: "Fick's Law").
In equilibrium condition free carriers set up themselves in such a way to built up a spatial charge [tex]\rho[/tex][tex]\neq[/tex]0. This charge is due to fixed ionized doping impurities not compensated by free carriers distribution and generates an internal electric field contrasting diffusional current.
This internal field is described by a potential function related to it by E(r)=-grad[tex]\varphi[/tex](r).
If you solve Schrodinger's Equation for stationary state of one free particle with this potential term, you will obtain band structure of the inhomogeneous system.
The solution is very simple if the variations of [tex]\varphi[/tex](r) potential are observable on length higher than primitive cell dimension.
The eigenvalues are then translated by a quantity dependent of position E(r)=E(0)-e[tex]\varphi[/tex](r); where E(0) is the usual parabolic solution of Schrodinger's equation in a homogeneous system that you can display in k-space.
In a homogeneous system [tex]\varphi[/tex]=cost and E(r)=E(0) everywhere, the bands are then flat; however the bands are bent. It' s important to note that differences on energy are not affected by the translation and so the energy gap.

Please correct my grammatical mistakes !

Thanks.
 
  • #3
One could transform the k-space picture into x-space picture, by a Fourier transform, but to my understanding one uses a combined view for the actual picture.
Basically one assumes the k-space picture at individual small localised cells, but these regions have to be large enough for k-space description to make sense (band diagrams are used for homogeneuos space only)
A more simplyfied approach is to take the lowest free energy positions in the band picture and say that dispersion E(k) corresponds to spaces for free particles with momentum k.
That's sort of the cell view described above.
 
  • #4
Yes, your interpretation - assuming k-space picture at small cells - is what I expected. But I think it is not easy to put this into a consistent mathematical form? I tried to interpred it along the same way as you do in accoustics when going from Fourier transform to "windowed Fourier transform". This also goes from a "fourier domain" to kind of a "fourier picture attached to each point of time". But I didn´t succeed in adapting this procedure to the band diagrams.
 

1. What is the difference between band diagrams in real space and reciprocal space?

The band diagram in real space is a representation of the energy levels of electrons in a material as a function of their position in real space. On the other hand, the band diagram in reciprocal space shows the energy levels of electrons as a function of their momentum in reciprocal space. In simpler terms, real space refers to physical space while reciprocal space refers to the space of wave vectors.

2. How are band diagrams in real space and reciprocal space related?

Band diagrams in real space and reciprocal space are related through Fourier transform. The band diagram in real space can be transformed into the band diagram in reciprocal space, and vice versa. This relationship allows scientists to study the properties of materials in both real and reciprocal space, providing a more comprehensive understanding of their electronic structure.

3. What information can we obtain from band diagrams in real space and reciprocal space?

Band diagrams in real space and reciprocal space provide valuable information about the electronic structure of materials. They can help determine the energy levels of electrons, their momentum, and the band gaps in a material. This information is crucial in understanding the electrical, optical, and magnetic properties of a material, which is essential for various technological applications.

4. How are band diagrams in real space and reciprocal space used in research?

Band diagrams in real space and reciprocal space are extensively used in research, particularly in the field of condensed matter physics. They are used to study the electronic structure of materials, investigate the properties of semiconductors, and understand the behavior of electrons in different materials. Band diagrams also play a crucial role in the design and development of new electronic and optoelectronic devices.

5. Are there any limitations to using band diagrams in real space and reciprocal space?

While band diagrams in real space and reciprocal space are powerful tools in studying the electronic structure of materials, they have some limitations. For instance, they do not account for the effects of temperature and impurities, which can significantly impact the behavior of electrons in a material. Additionally, these diagrams may not accurately represent the electronic structure of complex materials, making it challenging to interpret their properties accurately.

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