Graphene Energy Dispersion and Density of States: Understanding the Relationship

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SUMMARY

This discussion focuses on the properties of 2D graphene, specifically the determination of the number of nearest neighbors in a primitive cell and the calculation of the density of states (DOS). The number of nearest neighbors in graphene is established as 3. The linear energy dispersion near the Fermi level is described by the equation E=(hbar)vF|k|, leading to the density of states formula g(E)=E/2π(hbar)²vF². The discussion emphasizes the need for algebraic proof to connect the total number of states to the integral involving k(E) near the Dirac points.

PREREQUISITES
  • Understanding of 2D materials and their properties
  • Familiarity with quantum mechanics concepts, particularly energy dispersion
  • Knowledge of density of states calculations
  • Proficiency in mathematical integration techniques
NEXT STEPS
  • Study the derivation of the density of states in 2D systems
  • Explore the concept of Dirac points in graphene
  • Learn about the implications of linear energy dispersion in materials
  • Investigate the mathematical proof of the relation N=(A/2π)∫[between 0 and k(E)] dkk
USEFUL FOR

Students and researchers in condensed matter physics, materials science, and quantum mechanics, particularly those focusing on graphene and its electronic properties.

peripatein
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Hi,

Homework Statement


I have two questions, in fact, both involving 2D graphene:
(1) How may I determine the number of nearest neighbours in a primitive cell of graphene?
(2) Given that graphene has linear energy dispersion near the fermi level and the dispersion is given by E=(hbar)vF|k|, I would like to determine the density of states. I think it is equal to g(E)=E/2π(hbar)<sub>2</sub>v<sub>F</sub><sup>2</sup>, but how may I show that?<br /> <br /> I&#039;d appreciate your help.<br /> <br /> <br /> <h2>Homework Equations</h2><br /> <br /> <br /> <br /> <h2>The Attempt at a Solution</h2>
 
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I suppose I will aid in getting the ball rolling with this. There are a number of ways that you can approach this problem, however, it is rather difficult to help you without a gauge on your level of quantum mechanics.
 
Hi,
I believe I have managed through most of this. I have found the number of near neighbours to be 3. I even managed to nearly prove the required relation. However, what I am lacking is an explanation, or an algebraic proof, why N=total number of states=(A/2π)∗∫[between 0 and k(E)] dkk in the vicinity of the dirac points?
Also, how may I show that the units of g(E) in the above expression (my first post) are number of states per area per energy?
 

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