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I'm trying to solve this problem:

A GaAs LED is at $ 300 K $ when the current density is 100 $A/cm^2$ . The width of the active region is $ 1 µm $. Assume that, for that current, we are in the strong injection regime, and that the bimolecular coefficient is $10^{-10} cm^3 /s $. Calculate the cut-off frequency for that current density.

I know the answer is 4 MHz.

I know that the cut-off frequency relates do the carrier's lifetime.

$$f_c=\frac{1}{2 \pi \tau}$$

Right, so now I need to calculate the carrier's lifetime. I know that since this is strong injection we would have:

$$\tau=\frac{1}{B \Delta n_0}$$

where B is the bimolecular coefficient. So now I need to calculate $ \Delta n_0 $.

My question now is how can I use the temperature, the current density and the width of the active region to calculate $ \Delta n_0 $? I feel that I might need some parameters of GaAs at 300 K, but I'm having trouble understanding what parameters.

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