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The geometry of my semiconductor device is given below. The blue regions are gold, the grey ones - gallium arsenide (n-doped to $2.9 \times 10^{15} \mathrm{cm^{-3}}$). The dimensions are μm, i.e. it is 500μm wide, the bottom GaAs is 400μm thick, then a layer of gold follows, then a 15μm layer of GaAs, and a 5μm layer of Au on the top: geometry

The thinner strip of GaAs is a laser medium where radiation of 3THz is emitted. I would like to investigate the effect of the thickness of the thin layer of gold on the optical mode profile. I simulated the profile for two thicknesses: 20nm and 200nm. Below are the plots of the normalised E field, first 2D and then a cross section along x=0 line. Arc length is simply the y axis of the first plot.

20nm 20nm 2d 20nm 1d

200nm 200nm 2d 200nm 1d

In the first (20nm) case, |E| decreases from 450V/m to 90V/m (5x). In the second (200nm) case, |E| decreases from 2800V/m to 10V/m (280x).

The properties of the materials are as follows (Drude model for GaAs, literature for Au):

+----------+--------+---------+
| Material | Im(ε)  | Re(ε)   |
+----------+--------+---------+
| Au       | 1.43e5 | -5.11e4 |
| GaAs     | -0.17  | 12.98   |
+----------+--------+---------+

The field should decrease as $\exp(-\alpha z)$ (Beer's law), where $\frac{1}{\alpha} = \delta \approx 50\mathrm{nm}$ is the skin depth for 3THz in gold - this is calculated from: $$\alpha = \frac{1}{\delta} = \frac{\kappa \omega}{c}$$ where $\kappa$ is the known value for $Im(\tilde{n}) = 319$ (Ordal et al., http://dx.doi.org/10.1364/AO.26.000744). Hence: $$E_2(20nm) = 450 \exp(-20/50) \approx 300$$ $$E_2(200nm) = 2800 \exp(-200/50) \approx 50$$

The first result is 3x higher and the second 5x higher than I see in the simulation.

To check this, I simulated the system for different thicknesses of the gold layer, and fitted the data to the equation: $$\frac{|E_2|}{|E_1|} = A \exp \left(-\frac{z}{\delta} \right)$$

I got the following plot: fit

And the fit yields these parameters: $$\frac{|E_2|}{|E_1|} = 0.32 \exp \left(-\frac{z}{41\mathrm{nm}} \right)$$

So $\delta=41\mathrm{nm}$ is not too far from the calculated value. What I am trying to understand now is:

Where does the factor of 0.32 come from? It obviously matters much more for lower thickness of gold, which I am interested in. Does it have something to do with the normalisation? Boundary conditions?

I have realised that I should be looking at $\mathrm{E_y}$ rather than $\mathrm{|E|}$, but I checked it and the values look the same.

COMSOL FEM software was used to visualise the problem below. The question is related - albeit different - to this one: The skin effect and the reflectivity of gold

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  • $\begingroup$ what is a mode in this case? Do you have Dirichlet boundary conditions somewhere or does the software simulate some leaky situation? $\endgroup$ – Wolpertinger Apr 6 '17 at 9:38
  • $\begingroup$ All external boundaries are scattering (absorbing). $\endgroup$ – alkamid Apr 6 '17 at 10:29
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I think your refractive index for gold is not correct. I just did a quick look-up at refractiveindex.info for 100 µm, using published data by Ordal et al (which I usually use for a first guess).

Seems, your real part is almost there, but you're missing the imaginary part completely. Yet, this is significant, since it determines the losses of your gold layer. From the numbers, I would guess that the gold absorbs 2/3 of the light, which leaves you with the pre-factor of 0.32 to Beer's law.

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  • $\begingroup$ What's the value that you got? I double checked and $\kappa = 319$, both on refractiveindex.info and in Ordal's paper. $\endgroup$ – alkamid Mar 17 '16 at 18:28
  • $\begingroup$ You are using $ Im(n) $ to determine the skin depth. Are you also using it to determine optical losses in the Au layer? In your table, you list an imaginary value of 0 for Au. $\endgroup$ – engineer Mar 21 '16 at 18:14
  • $\begingroup$ Sorry, that 0 in the table was confusing — I was using a loss model in COMSOL which only required one part of $\epsilon$ and conductivity, as they are dependent. $\endgroup$ – alkamid Apr 6 '17 at 8:57
  • $\begingroup$ I edited the table in my original question. When I tried running the model with those parameters, it didn't change significantly. I am not sure I understand your comment about gold absorption — why should there be additional effects, not described by Beer's law? (i.e. why the pre-factor of 0.32) $\endgroup$ – alkamid Apr 6 '17 at 9:07
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I think the assumption that I should see decay according to Beer's law is wrong. It would apply if I sent a wave through a thin layer of gold and looked at the transmission. In my original model though, I was dealing with a wave confined in a sub-wavelength cavity, so the electric field norm was a sum of reflections between the two layers of metal.

This is not to say there is nothing spurious with the results I got, but it is probably not the right way of checking them.

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