1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Energy gap between brillouin zones?

  1. Jun 12, 2015 #1
    1. The problem statement, all variables and given/known data

    Consider a monovalent 2D crystal with a rectangular lattice constants ##a## and ##b##. Find expressions for the fermi energy and fermi wavevector in 2D. Show that the fermi surface extends beyond first zone if ## 2a > b\pi##. If the crystal is now divalent, estimate the energy gap between the first and second brillouin zone.

    2. Relevant equations


    3. The attempt at a solution

    Part(a)
    I found the density of states to be ## g(E) dE= \frac{1}{\pi} \frac{m}{\hbar^2} dE##. Fermi energy is then found to be ##E_F = \frac{n\pi \hbar^2}{m} ##. Wavevector is also found to be ##k_F = (2n\pi)^{\frac{1}{2}}##. Since it atom is monovalent, ##n = \frac{1}{ab}##. The fermi energy and wavevector thus becomes ##E_F = \frac{\pi \hbar^2}{mab}## and ##k_F = \left( \frac{2\pi}{ab} \right)^{\frac{1}{2}}##.

    How do I continue and show the relation ##2a > b\pi##?
     
  2. jcsd
  3. Jun 13, 2015 #2
    When ## a=b ## then ## 1 < \pi/2 ##.
     
  4. Jun 13, 2015 #3
    How did you get that?
     
  5. Jun 13, 2015 #4
    1st zone is between ##\pm \pi/2 ##.
     
  6. Jun 13, 2015 #5
    I just solved the first part. The brillouin zone exceeds when
    [tex]k > \frac{\pi}{a}[/tex]
    [tex]\sqrt{\frac{2\pi}{ab}} = \frac{\pi}{a}[/tex]

    For the second part,
    is the gap simply
    [tex]\frac{\hbar^2}{2m}( k_x^2 - k_y^2) = \frac{\hbar^2 \pi^2}{2m} \left( \frac{1}{a^2} - \frac{1}{b}^2 \right) [/tex]
     
  7. Jun 13, 2015 #6
    Ok is the same. I use numerics. Check values of ##a,b## for left part be smaller than right.
     
  8. Jun 13, 2015 #7
    It's given that ##a>b##. Is my part (b) right?
     
  9. Jun 13, 2015 #8
    inverce a,b roles. the answer is indepented to which is a an which is b
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Energy gap between brillouin zones?
  1. Brillouin zone (Replies: 3)

  2. 1st Brillouin zone (Replies: 0)

  3. Brillouin Zone Size (Replies: 3)

Loading...