# Kronig-Penney Model: Plotting Energy Bands in Excel

• Fez07
In summary, you need to set up your spreadsheet to plot lines and the RHS vs E. The areas where the values are greater than 1 are solutions with bands.f

#### 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.

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.

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.

Thanks

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.

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

Thanks

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