Calculate the generation rate of electron-hole pairs given an incident intensity of photons.

Consider gallium arsenide at $T=300 \mathrm{~K}$ . Assume the photon intensity at a particular point is $I_{\nu}(x)=0.05 \mathrm{~W} / \mathrm{cm}^{2}$ at a wavelength of $\lambda=0.75 \mu \mathrm{m}$ . This intensity is typical of sunlight, for example.

Question

Calculate the generation rate of electron-hole pairs given an incident intensity of photons.

Consider gallium arsenide at [latex]T=300 \mathrm{~K}[/latex]. Assume the photon intensity at a particular point is [latex]I_{ u}(x)=0.05 \mathrm{~W} / \mathrm{cm}^{2}[/latex] at a wavelength of [latex]\lambda=0.75 \mu \mathrm{m}[/latex]. This intensity is typical of sunlight, for example.

Accepted Answer

The absorption coefficient for gallium arsenide at this wavelength is [latex]\alpha \approx 0.9 imes 10^{4} \mathrm{~cm}^{-1}[/latex]. The photon energy, using Equation (14.1), is

[latex]\lambda=\frac{c}{ u}=\frac{h c}{E}=\frac{1.24}{E} \mu \mathrm{m}[/latex] (14.1)

[latex]E=h u=\frac{1.24}{0.75}=1.65 \mathrm{eV}[/latex]

Then, from Equation (14.6) and including the conversion factor between joules and [latex]\mathrm{eV}[/latex], we have, for a unity efficiency factor,

[latex]\begin{aligned} g^{\prime}=\frac{\alpha I_{ u}(x)}{h u}=\frac{\left(0.9 imes 10^{4} ight)(0.05)}{\left(1.6 imes 10^{-19} ight)(1.65)}=1.70 imes 10^{21} \mathrm{~cm}^{-3}-\mathrm{s}^{-1}\end{aligned}[/latex]

If the incident photon intensity is a steady-state intensity, then, from Chapter 6 , the steadystate excess carrier concentration is [latex]\delta n=g^{\prime} au[/latex], where [latex] au[/latex] is the excess minority carrier lifetime. If [latex] au=10^{-7} \mathrm{~s}[/latex], for example, then

[latex]\delta n=\left(1.70 imes 10^{21} ight)\left(10^{-7} ight)=1.70 imes 10^{14} \mathrm{~cm}^{-3}[/latex]

Comment

This example gives an indication of the magnitude of the electron-hole generation rate and the magnitude of the excess carrier concentration. Obviously, as the photon intensity decreases with distance in the semiconductor, the generation rate also decreases.

Question 14.2: Calculate the generation rate of electron–hole pairs given a...

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