# Calculating the Dielectric Function of metals - Units trouble

1. ### zellwwf

33
Hello everyone here!

See i got my hands on this paper:
Optical properties of metallic films for vertical-cavity optoelectronic devices
by Rakic et al.

Now, i am writing the function in matlab to calculate the permittivity based on the Lorentz-Drude model mentioned in the paper.
The paper however doesn't help with the units. It's my fault i know.. i should know my units. But i've tried a lot of stuff and nothing seems to work.. i am getting wierd graphs that don't look anything like the ones in the paper. Not even close. I rewrote the matlab function a number of times and i don't think it's that. i think its in the units.

Here is what i need help with:
I believe that ε(w) is unitless. Correct me here if i am wrong.
(i will be using silver as an example here)
Plasma Frequency of silver is 9.01 (see ref) eV.
$\omega_p* \hbar = 9.01 eV$
And if i am not wrong in formulas (1) (2) (3), he references [
itex]\omega_p[/itex], so i automatically divided 9.01 by Hbar (in Ev.S units), but that would give inconsistent results with a Unitless permittivity. It would give units of: $s^{-2}.eV^{-2}$ for the permittivity function he mentions.

so let's check what have i done:
1) Feed 9.01 into the formula where wp is. Wrong Graph.
2) Feed 9.01/Hbar into the formula where wp is. Wrong Graph again.

Can someone tell me what are the units for the formula (1,2,3) in the ref? and what kind of normalizations or unit conversions i have to do in order to put the data from Tables 1, 2 (ref) to make it work?
If i am getting anything wrong, tell me please.

2. ### DrDu

4,094
Yes, the ##\epsilon_r## he defines is unitless.
Apart from this, I don't understand your problem: both ##\omega## and ##\Gamma## have the same units as ##\Omega_p## so that the whole quotient is unitless, as it should be.
The question is rather: What units are you using for ##\omega##?

3. ### zellwwf

33
I am using eV, in other words.. i am running a loop from 0.01 ev to 1 ev. Feeding 0.01 to the function.
But $\Omega_p$ is defined as $\sqrt{f}\omega_p$, where omega is in Hz, not eV, as i noted in table 2, in table 2 he says that the value 9.01 is hbar times plasma freq.

Should i divide by hbar? but still that gave me a wrong graph.
I you need, i can send you my results.

If i divided by hbar, that means Omega is not in eV anymore, it's in Hz, which means it the units won't cancel out.

Last edited: Aug 19, 2013
4. ### DrDu

4,094
No matter what you think he may write. If your ##\omega## is in eV, ##\Omega_p## has to be in eV, too.

5. ### zellwwf

33
I agree... so i shouldnt' divide by hbar?