Question

Graphene is a single graphitic layer of carbon atoms, which has the remarkable feature that electrons in graphene behave as two dimensional massless relativistic fermions with a "speed of light, graphene 《 c. Andre Geim and Konstantin Novoselov were awarded the 2010 Nobel Prize in Physics "for groundbreaking experiments regarding the two-dimensional material graphene". a) ) Calculate the density of states as a function of energy for massless relativistic electrons in graphene with for a system of size L (note that the degeneracy factor is g 4) ii) Calculate the density of states as a function of energy for non-relativistic two dimensional electrons in 2 dimensions with spin degeneracy 9-2 for a system of size L b) Use your results from part a) to calculate the Fermi energy eF and average energy U at zero temperature when there are N particles for both non-relativistic electrons and electrons in graphene. Express your answers for U in the form U αΝ6p and determine o. c) Find the chemical potential at all temperatures in the non-relativistic case. Show that your expression gives the correct behaviour at low and high temperatures.

Answer 1

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mO 2m 4兀 2丌 丹m im 2 方2

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