# Estimating the Ratio of Excited Electrons in GaAs at 300 K and 500 K ## Step 1: **Identify the Relevant Formula** The number of thermally excited electrons (intrinsic carrier concentration, \( n_i \)) in a semiconductor is given by: \[ n_i \propto \exp\left(-\frac{E_g}{2kT}\right) \] where: - \( E_g \) = band gap energy = **1.42 eV** - \( k \) = Boltzmann constant = **8.617 × 10^{-5} \text{ eV/K}** - \( T \) = temperature in Kelvin To find the ratio at two temperatures (300 K and 500 K): \[ \text{Ratio} = \frac{n_{i,500}}{n_{i,300}} = \frac{\exp\left(-\frac{E_g}{2k \cdot 500}\right)}{\exp\left(-\frac{E_g}{2k \cdot 300}\right)} \] This simplifies to: \[ \text{Ratio} = \exp\left(\frac{E_g}{2k} \left(\frac{1}{300} - \frac{1}{500}\right)\right) \] **Explanation:** The intrinsic carrier concentration increases exponentially with temperature, controlled by the band gap and Boltzmann factor. --- ## Step 2: **Plug in the Values and Calculate** First, compute the factor inside the exponential: - \( \frac{1}{300} - \frac{1}{500} = \frac{5 - 3}{150} = \frac{2}{150} = .00133 \) - \( \frac{E_g}{2k} = \frac{1.42}{2 \times 8.617 \times 10^{-5}} = \frac{1.42}{1.7234 \times 10^{-4}} \approx 8243 \) Now, plug into the ratio: \[ \text{Ratio} = \exp(8243 \times .00133) = \exp(10.95) \approx 56,\!900 \] **Explanation:** This calculation shows how much more likely electrons are to be thermally excited at 500 K compared to 300 K due to the exponential dependence on \( \frac{1}{T} \). --- ## **Final Answer** \[ \boxed{\text{The ratio of the number of excited electrons at 500\,K to 300\,K for GaAs (E_g = 1.42\,eV) is approximately 56,\!900.}} \]

# Estimating the Ratio of Excited Electrons in GaAs at 300 K and 500 K ## Step 1: **Identify the Relevant Formula** The number of thermally excited electrons (intrinsic carrier concentration, \( n_i \)) in a semiconductor is given by: \[ n_i \propto \exp\left(-\frac{E_g}{2kT}\right) \] where: - \( E_g \) = band gap energy = **1.42 eV** - \( k \) = Boltzmann constant = **8.617 × 10^{-5} \text{ eV/K}** - \( T \) = temperature in Kelvin To find the ratio at two temperatures (300 K and 500 K): \[ \text{Ratio} = \frac{n_{i,500}}{n_{i,300}} = \frac{\exp\left(-\frac{E_g}{2k \cdot 500}\right)}{\exp\left(-\frac{E_g}{2k \cdot 300}\right)} \] This simplifies to: \[ \text{Ratio} = \exp\left(\frac{E_g}{2k} \left(\frac{1}{300} - \frac{1}{500}\right)\right) \] **Explanation:** This formula reveals that the intrinsic carrier concentration increases exponentially with temperature, influenced by the band gap energy and the Boltzmann factor. --- ## Step 2: **Plug in the Values and Calculate** First, compute the factor inside the exponential: - Calculate \( \frac{1}{300} - \frac{1}{500} \): \[ \frac{1}{300} = 0.003333, \quad \frac{1}{500} = 0.002 \] \[ \frac{1}{300} - \frac{1}{500} = 0.003333 - 0.002 = 0.001333 \] - Calculate \( \frac{E_g}{2k} \): \[ \frac{E_g}{2k} = \frac{1.42}{2 \times 8.617 \times 10^{-5}} \approx \frac{1.42}{1.7234 \times 10^{-4}} \approx 8243 \] Now, plug into the ratio: \[ \text{Ratio} = \exp(8243 \times 0.001333) = \exp(10.99) \approx 59,000 \] **Explanation:** The calculations demonstrate a significant increase in the number of thermally excited electrons at 500 K compared to 300 K, highlighting the strong dependence on temperature. --- ## **Final Answer** \[ \boxed{\text{The ratio of the number of excited electrons at 500 K to 300 K for GaAs (E_g = 1.42 eV) is approximately 59,000.}} \]