Semiconductors — Question 6

Question 6

A transient conductivity measurement determines the recombination lifetime

After a weak light pulse, a uniform semiconductor follows $\Delta\sigma(t)=\Delta\sigma_0e^{-t/\tau}$. Measurements at $t_1=1.00\,\mu\mathrm s$ and $t_2=5.00\,\mu\mathrm s$ have ratio $\Delta\sigma(t_1)/\Delta\sigma(t_2)=4$.

1. Infer the lifetime without knowing the initial pulse amplitude.

2. For initial excess density $\Delta n_0=10^{16}\,\mathrm{m^{-3}}$, find the number of recombined pairs per volume by $t=\tau$.

Question 6 - Solution

A transient conductivity measurement determines the recombination lifetime

(a) Method and calculation

$$\ln\frac{\Delta\sigma(t_1)}{\Delta\sigma(t_2)}=(t_2-t_1)/\tau.$$$$\boxed{\tau=4.00\,\mu\mathrm s/\ln4=2.885\,\mu\mathrm s}.$$

(b) Interpretation and further calculation

$$N_{\mathrm{rec}}/V=\int_0^t\frac{\Delta n_0}{\tau}e^{-t\prime/\tau}dt\prime=\Delta n_0(1-e^{-t/\tau}).$$$$\boxed{N_{\mathrm{rec}}(\tau)/V=6.321\times10^{15}\,\mathrm{m^{-3}}}.$$

Physical check

The pair count equals the loss of either excess species, not their sum. A curved semilog decay would indicate that one constant lifetime is insufficient.