Pursuant to the limitation that it only applies to countries having made an agreement with Eurocontrol for collecting en-route charges (all European countries, I presume), the calculation formula can be found here:
https://www.eurocontrol.int/sites/de...of-payment.pdf
And the administrative context is described here:
https://www.eurocontrol.int/articles...-route-charges
In short, the factor that brings in the weight of the plane is:
p = 0 , if the MTOW of the aircraft is less than 2 metric tons,
p = square root of (MTOW/50) , with MTOW measured in metric tons, in all other standard cases
So it is neither a linear nor a step up factor.
For an aircraft whose MTOW is exactly 2 tons, the factor p = 0.2
12.5 tons, p = 0.5
50 tons, p = 1
200 tons, p = 2