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Homework Help: Kronig-Penney Model

  1. Mar 27, 2007 #1
    Hi there,
    I want to plot a graph of the right hand side of the equation VS the energy, E, in eV for a periodic array of square wells each of depth 300eV, width 0.05nm, and period 0.1nm using SI units and MICROSOFT EXCEL spreadsheet to determine the allowed energy bands and band gaps:

    cos(kp)=F=cos(α(p-w))cosh(βw)+((β squared - α squared)/2αβ))sin(α(p-w))sinh(βw)

    α = (1/h bar)* square root(2mE)

    β = (1/h bar)* square root(2m(V-E))

    w = width of barrier separating two adjacent potential wells
    p = period
    m = mass of an electron
    V = well depth

    Solutions are only obtained if F is between -1 and 1.

    I have no idea how to do this so any help would be much appreciated.

    Thanks in advance.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 28, 2007 #2

    Dr Transport

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    First, you need to set up your spreadsheet with the variable quantities and constants. We insist that you attempt a solution before we help you.
  4. Mar 30, 2007 #3
    Thanks for replying. I set up the EXCEL spread sheet with columns of:

    V = 300*1.6*10^-19 (energy in J)
    p = 0.1*10^-9 (period in m)
    m = 9.1094*10^-31 (mass of the electron in kg)
    h_bar = 1.05*10^-34
    w = 0.05*10^-9 (width of separating barrier in m)

    E = alpha^2*h_bar^2 / (2*m)
    β = (1/h_bar)* square root(2m(V-E))

    and the RHS of the equation.

    But i don't know what to do from here.
    Please help me.

  5. Mar 31, 2007 #4

    Dr Transport

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    So if you've set up your columns, plot 3 lines, [itex] \pm 1 [/itex] and the RHS vs [itex] E [/itex]. The areas where the values are greater than 1 are solutions where there are bands. Later I'll plot up your solutions and see what I get.
  6. Apr 2, 2007 #5
    Hi again,
    I still can't figure it out. I'm getting a "Circular Reference" error between the alpha and E values..?


  7. Apr 3, 2007 #6
    Hi there, it's me again:

    I've figured out what i was doing wrong and gotten the solution.

    Thanks for the help. I couldn't have done it otherwise.

    Two thumbs up.

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