The Relationship Between Dielectric Function and Joint Density of States

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Discussion Overview

The discussion centers on the relationship between the imaginary part of the dielectric function (epsilon_2) and the joint density of states (JDOS). Participants explore theoretical aspects, particularly in the context of isotropic versus anisotropic materials, and the implications for transitions in crystal structures.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether the amplitude of epsilon_2 is directly proportional to JDOS or if JDOS is a derivative of epsilon_2.
  • Another participant states that epsilon_2 is almost directly proportional to JDOS, with exact proportionality depending on the independence of the matrix element for transitions in k-space.
  • A participant inquires about the changes in the integral for epsilon in anisotropic materials, suggesting that the matrix element's dependence on the polarization vector differs from isotropic materials.
  • It is noted that the coupling between conduction and valence bands is anisotropic and that the coupling between valence band states must also be considered.
  • A later reply reiterates the relationship between epsilon_2 and JDOS, introducing a factor of 1/E^2 in the epsilon_2 equation and questioning the proportionality of JDOS to E^2*epsilon_2.

Areas of Agreement / Disagreement

Participants express varying views on the relationship between epsilon_2 and JDOS, with some suggesting a direct proportionality under certain conditions while others introduce complexities related to anisotropic materials and matrix elements. The discussion remains unresolved regarding the exact nature of these relationships.

Contextual Notes

Participants mention dependencies on the matrix element's position in k-space and the effects of anisotropy, indicating that assumptions about isotropy may limit the applicability of certain formulas.

jet10
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Hi. I have been looking at some lecture notes. What is not so clear for me is, how the imaginary part of the dielectric function is related to the joint density of states. Is the "amplitude" of the epsilon2 directly proportional to JDOS? or is JDOS some kind of derivative of epsilon2?

Thanks
 
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Epsilon_2 is *almost* directly proportional to the JDoS. It is exactly proportional if the matrix element for the transition is independent of the position in k-space on the surface that defines the energetically allowed transition. For most purposes in crystals, the matrix element is only weakly dependent, and people like to just move it outside of the integral and replace it with an averaged matrix element.
 
Thanks for your clear answer. Just one more question. I see that the formula for Epsilon in books are normally given for isotropic material. What changes in the integral of the formula if we want to know Epsilon in a certain direction for anisotropic material? There is a polarisation vector e in the matrix element for the transition <c|e.p|v>. I guess that for anisotropic material, the matrix element will depend on which e or which direction I take, whereas for isotropic material, it doesn't matter. Is this right?
 
Yes, the coupling between the conduction and valence band will be anisotropic. You also have to include the coupling between the valence band states.
 
Ok. Thanks very much for the help!
 
genneth said:
Epsilon_2 is *almost* directly proportional to the JDoS. It is exactly proportional if the matrix element for the transition is independent of the position in k-space on the surface that defines the energetically allowed transition. For most purposes in crystals, the matrix element is only weakly dependent, and people like to just move it outside of the integral and replace it with an averaged matrix element.

I noticed that there is a factor of 1/E^2 in the \varepsilon_2 equation. Since \varepsilon_2 is dependent on E, isn't the JDOS rather *almost* proportional to E^2\varepsilon_2?
 

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