Refractive Index change with Wavelength & Carrier Concentration

[rbert15]

I need to calculate the refractive index of a semiconductor material over a wavelength range (1×10^-5m - 1×10^-9m) and with different values of electron and hole carrier concentrations (i.e. n/p doped).

I found this equation that relates those parameters:

n+ik = √ [ (ε_m - [(e²/ω²)*((n₀/m*_e)+(p₀/m*_h))]) / ε₀ ]

The values ε₀ and e are known constants. For each semiconductor material I'm interested in, the values of electron and hole concentrations (n₀ and p₀) and the effective electron and hole masses (m*_e and m*_h) have been found, as well as the value of the dielectric constant/relative permittivity (ε_m), all via literature (http://www.ioffe.ru/SVA/NSM/)

The problem is that whilst the equation is giving me results, the value of the refractive index (n+k) is not varying as much as it should when the wavelength is changed. For example for InAs it should change from about 4 to 1, whereas it is staying almost constant at 3.5 regardless of wavelength.

The frequency (ω) in the equation is calculated by simply ω=c/λ and this value (and the subsequent term) is so small compared to ε_m and ε₀ that it doesn't have much influence. Does ε_m have a wavelength/frequency dependence I'm not accounting for? I can't see any other variables that would have λ dependence to increase λ effect on n calculation.

 

[blue_leaf77]

A very good idea is to recheck the reference where you got that equation from. Second, the LHS is complex while the RHS seems to always be real unless one also takes the absorption in relative permittivity into account.

 

[rbert15]

The equation is from the Plasma-Drude model as shown here http://docs.lumerical.com/en/index.html?ref_sim_obj_charge_to_index_conversion.html but I cannot source the reference "Henry, C. H.; Logan, R. A.; Bertness, K. A. Journal of Applied Physics, vol. 52, (1981), p. 4457-4461" to view the original work.

I have checked the units match across the equation.

The RHS should be complex. There must be more to the ε_r value than just (number*ε₀) then - but I don't know what/how to include this.

 

[blue_leaf77]

Ah yes, that is free carrier contribution to the refractive index. According to Drude model, the pure plasma (no bound electrons) system yields the following expression for refractive index
$$ n(\omega) = \sqrt{1-\frac{e^2N}{\epsilon_0 m \omega^2}} $$
Note that the second term is somehow similar to the corresponding one in the above equation. When all contributions present such as those related to bound electrons, the corresponding susceptibility must also be included which is $\epsilon_m$. In other words $\epsilon_m$ must also be a function of frequency.

 

[rbert15]

So, in first equation how is ε_m term modified to include λ dependence?

 

[blue_leaf77]

The link you gave there says that $\epsilon_m$ corresponds to permittivity of unperturbed system, I'm not a pro in semiconductor but I guess it is closely related to the host material before being dopped.

 

[nasu]

I would check the reference for the range of validity of the formula. You want to use is over a 4 orders of magnitude range of wavelengths, from infrared to x-rays.
I would be surprised if it gives accurate results for both x-rays and infrared, in a single formula.

 

[rbert15]

The values of n over the λ range as calculated using that formula should be valid as that is the method used by Lumerical (where I found the formula) to obtain the "n vs λ" values in their materials database. This is how I know whether my calculations are correct, as I can compare values of n for example with undoped InAs (with intrinsic carrier concentrations) before then moving on to include doping.

Through some further reading I find that "at low frequency the expression for ε_r reduces to static field case" - this is why the results at the λ=10^-5 are close in agreement but way off at the λ=10^-9 end of the wavelength range.

I was using just ε_r=*number for static field case* whereas I should be using ε_r=1+χ_e to account for electric susceptibility - which has a wavelength dependence.

χ_e=(N*e²)/(ε₀ω₀²m) but don't know what are the definitions of N or ω₀² or m. I thought N is number of atoms per cm³, λ₀ is the wavelength, and m is electron mass, but that doesn't give the overall value required.

I think if I can include χ_e that will solve the issue.

 

[blue_leaf77]

Note that Drude model is based on classical calculation.
As long as your reference really does mention the range of validity, only then can you be confident with your calculation.

I should be using εr=1+χe to account for electric susceptibility - which has a wavelength dependence.

This is the point where Sellmeier's equation comes into play. Don't bother trying to find a closed form of the susceptibility as a function of wavelength for that large range. And also note that Sellmeier's equation for most material are typically defined with high degree of accuracy only within about 0.2 microns up to about 3 microns.

 

[nasu]

The values of n over the λ range as calculated using that formula should be valid as that is the method used by Lumerical (where I found the formula) to obtain the "n vs λ" values in their materials database. This is how I know whether my calculations are correct, as I can compare values of n for example with undoped InAs (with intrinsic carrier concentrations) before then moving on to include doping.

Do they give values of index of refraction for wavelengths in the nano meter range?

 

[rbert15]

The material database gives a list of wavelength (or frequency) vs refractive index (real and imag parts) only for the range 2x10^-5 to 5x10^-8 - so it doesn't go into nanometre range (which is interesting to note since a lot of simulations take place over the nm range).

I think perhaps I'm looking for a simple answer when there isn't one. I thought that for example for InAs ε_r=12.3 for static field case then since ε_r=1+χ_e then χ_e would have some simple λ dependence and be a value varying around 1-11.

Is it possible to establish a value for χ_e over smaller wavelength ranges, say MIR (5-20μm) or nm? I still don't know what all the parameters mean in the χ_e definition.

 

[blue_leaf77]

Is it possible to establish a value for χe over smaller wavelength ranges, say MIR (5-20μm) or nm?

As I have said, you should look for the Sellmeier equation for the material of interest. Example of this equation for InAs can be found in this site
http://refractiveindex.info/?shelf=main&book=InAs&page=Aspnes
Just inform yourself about the wavelength range and accuracy of data assumed in this website.

 

[rbert15]

That's a really useful webpage - thanks for the link. I see I can select "0.2-0.8um" range and it shows values of n and ε and states the n doping; or I can select "3-31um" which shows n but doesn't state the n doping (would assume it to be the same, intrinsic). There's an equation for n which includes λ but this doesn't include influence of the doping.

So, I'm still stuck on how to calculate refractive index n based on wavelength and n doping value.

 

[blue_leaf77]

If I were you I would try to calculate the overall refractive index including doping using the equation you found up there
n+ik = √ [ (εm - [(e2/ω2)*((n0/m*e)+(p0/m*h))]) / ε0 ]
with $\epsilon_m = (n_0 + ik_0)^2$, $n_0$ and $k_0$ being the values obtained from that webpage. Again this is purely my speculation, I have got no reference for its justification.

 

[rbert15]

I think that sounds like a good idea - to calculate ε_m using n₀ - and then input this into the first equation.