Cut-off frequency of a LED
After a bit of digging, I found this:
Equation 9.17 in this chapter
You can manipulate that equation as -
$$n=\sqrt{\frac{J}{edB}}$$
Given:
\$J=1000A/cm^2\$
\$B=10^{-10}cm^3/s\$
\$d=10^{-4}cm\$
\$e=1.6\times10^{-19}C\$
Therefore -
$$n=\sqrt{\frac{10^{36}}{1.6}}\approx\mathbf{0.791\times10^{18}/cm^3}$$
Calculate carrier lifetime -
$$\tau=\frac{1}{B \Delta n}\approx\mathbf{12.64 \text{ ns}}$$
Calculate cut-off frequency -
$$f_c=\frac{1}{2 \pi \tau}\approx\mathbf{12.6\text{ MHz}}$$