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Understanding excess carrier concentration is crucial when dealing with semiconductors influenced by an external stimulus like light. When a semiconductor like GaAs is exposed to light, additional electrons and holes are generated beyond what is thermally generated. These are known as excess carriers. In the given exercise, excess carriers are created at a rate given by \(g' = 4 \times 10^{21} \text{ cm}^{-3} \text{ s}^{-1}\). The concentration varies over time, starting from the moment the light is turned on (t = 0) and ends when it is turned off (t = 10^{-6} s).
During the exposure period, the number of excess carriers increases linearly with time: \(\Delta p(t) = 4 \times 10^{21} \cdot t\). Once the light source is off, these carriers begin to decay over time due to recombination, following an exponential decay law: \(\Delta p(t) = 4 \times 10^{15} \cdot e^{-(t-10^{-6})/(5 \times 10^{-8})}\). This decay demonstrates the inherent instability of excess carriers when no longer fueled by the external stimulus.

Minority carrier lifetime, denoted as \(\tau_{p0}\) in this exercise, is the average time a minority carrier, such as holes in an n-type semiconductor, can exist before recombination. It measures the endurance of carriers against annihilation under no external carrier generation conditions. In the context of the exercise, \(\tau_{p0} = 5 \times 10^{-8} \text{ s}\) determines how quickly the excess holes (minority carriers in n-type) recombine after the light source is switched off.
This attribute is fundamental in understanding carrier dynamics as it directly influences how long the semiconductor can retain its newly added charge carriers once the addition ceases. The exponentially decaying function \(e^{-(t-10^{-6})/\tau_{p0}}\) describes how minority carriers diminish over time post illumination, reminding us that lifetime affects the persistence of elevated carrier levels.

Conductivity, a key attribute describing how well a material can conduct electric current, is given by the equation: \[\sigma(t) = q \cdot (N_d \cdot \mu_n + \Delta p(t) \cdot \mu_p)\] where \(q\) is the charge of an electron, \(N_d\) is the donor concentration, \(\mu_n\) and \(\mu_p\) are mobilities of electrons and holes, respectively. For n-type GaAs, the conductivity changes as the concentration of excess carriers \(\Delta p(t)\) varies with time.
Initially, during the light exposure \(0 \leq t \leq 10^{-6}\), \(\Delta p(t)\) increases linearly contributing an additional component to the semiconductor's conductivity. The expression adapts for \(\Delta p(t) = 4 \times 10^{21} \cdot t\). When the light source is off, the decay in \(\Delta p(t)\) translates to a decrease in conductivity. By substituting \(\Delta p(t) = 4 \times 10^{15} \cdot e^{-(t-10^{-6})/(5 \times 10^{-8})}\) for \(t > 10^{-6}\), one can track conductivity as it returns towards its base level, emphasizing the delicate balance between generation and recombination rates in determining electrical characteristics.