Kronig-Penney Model

[Fez07]

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

[Dr Transport]

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.

[Fez07]

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.

Thanks

[Dr Transport]

So if you've set up your columns, plot 3 lines, \pm 1 and the RHS vs E . 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.

[Fez07]

Hi again,
I still can't figure it out. I'm getting a "Circular Reference" error between the alpha and E values..?

Thanks

Fez07

[Fez07]

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.

Fez07