Chapter 18: Problem 35

Estimate the temperature at which GaAs has an electrical conductivity of \(1.6 \times 10^{-3}\) \((\Omega-m)^{-1}\) assuming the temperature dependence for \(\sigma\) of Equation \(18.36 .\) The data shown in Table 18.3 might prove helpful.

Short Answer

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Question: Estimate the temperature at which Gallium Arsenide (GaAs) has an electrical conductivity of \(1.6 \times 10^{-3} (\Omega m)^{-1}\) using the given information about GaAs and its intrinsic carrier concentration, electron mobility, and hole mobility.

Step by step solution

Write the equation for electrical conductivity

We will use equation 18.36 for the temperature dependence of electrical conductivity, which is given by: \(\sigma = q(n_i [\mu_n + \mu_p])\) Here, \(\sigma\) is the electrical conductivity, \(q\) is the electronic charge (\(1.6\times10^{-19} C\)), \(n_i\) is the intrinsic carrier concentration, and \(\mu_n\), \(\mu_p\) are the electron and hole mobilities, respectively.

Find the intrinsic carrier concentration

From Table 18.3, we know the electron effective mass \((m_{e}^*)\) and hole effective mass \((m_{h}^*)\) for GaAs as \(0.067m_0\) and \(0.45m_0\), respectively, where \(m_0\) is the rest mass of an electron. We also know the lattice constant \((a)\) and the bandgap energy \((E_{g})\) as \(5.65\, Angstrom\) and \(1.52\, eV\), respectively. To calculate the intrinsic carrier concentration, we can use the formula: \(n_i = 2\left(\frac{2\pi k T}{h^2}\right)^{\frac{3}{2}}a^3\sqrt{m_e^*m_h^*}e^{\frac{-E_g}{2kT}}\) Here, \(k\) is the Boltzmann constant \((1.38\times10^{-23}\, J/K)\), \(T\) is the temperature, and \(h = 6.626\times10^{-34} Js\) is the Planck constant.

Find the electron and hole mobilities

We know that: \(\mu_n = \frac{e \tau_n}{m_{e}^*}\) and \(\mu_p = \frac{e \tau_p}{m_{h}^*}\) Here, \(\tau_n\) and \(\tau_p\) are the electron and hole mean free times, respectively. For GaAs, from Table 18.3, we have: \(\tau_n = 3.2 \times 10^{-14}s\) and \(\tau_p = 3.2 \times 10^{-14}s\) Now, we can calculate the electron and hole mobilities.

Find the temperature

We have the electrical conductivity (\(\sigma\)) given as \(1.6 \times 10^{-3} (\Omega m)^{-1}\). Using the equation from Step 1, we can write: \(1.6 \times 10^{-3} = q(n_i [\mu_n + \mu_p])\) Substitute the values of \(n_i\), \(\mu_n\), and \(\mu_p\) from Steps 2 and 3 into this equation and solve for \(T\). This will give us the temperature at which GaAs has an electrical conductivity of \(1.6 \times 10^{-3} (\Omega m)^{-1}\). Once the temperature has been found, the problem is solved.

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Estimate the temperature at which GaAs has an electrical conductivity of \(1.6 \times 10^{-3}\) \((\Omega-m)^{-1}\) assuming the temperature dependence for \(\sigma\) of Equation \(18.36 .\) The data shown in Table 18.3 might prove helpful.

Short Answer

Step by step solution

Write the equation for electrical conductivity

Find the intrinsic carrier concentration

Find the electron and hole mobilities

Find the temperature

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